363. Geometry problems with perpendicularities.

by Virgil Nicula, Dec 1, 2012, 12:37 PM

PP11. Let $\triangle ABC$ and for an interior point $P$ denote $\left\{\begin{array}{c} D\in AP\cap BC\\\ E\in BP\cap CA\\\ F\in CP\cap AB\end{array}\right|$ . Prove that $\frac{1}{PD}=\frac{1}{AD}+\frac{1}{BD}+\frac{1}{CD}\Longrightarrow DE\perp DF$ (2/4/2013).

Proof. Apply the Menelaus' theorem to the transversals: $\left\{\begin{array}{ccccc} \overline{BPE}/\triangle ADC\ : & \frac {BD}{BC}\cdot \frac {EC}{EA}\cdot \frac {PA}{PD}=1 & \implies & \frac {BD}{BC}\cdot\frac {PA}{PD}= \frac {EA}{EC} & (1)\\\\ \overline{CPF}/\triangle ADB\ : & \frac {CD}{CB}\cdot \frac {FB}{FA}\cdot \frac {PA}{PD}=1 & \implies & \frac {CD}{CB}\cdot\frac {PA}{PD}= \frac {FA}{FB} & (2)\end{array}\right|$ . Therefore, $\frac{1}{PD}=\frac{1}{AD}+\frac{1}{BD}+\frac{1}{CD}$ $\iff$

$\frac{1}{PD}-\frac{1}{AD}=\frac{1}{BD}+\frac{1}{CD}$ $\iff$ $\frac{PA}{PD\cdot AD}=\frac{BC}{BD\cdot CD}$ $\iff$ $\left\{\begin{array}{ccccc} \frac {BD}{BC}\cdot \frac{PA}{PD}=\frac {DA}{DC} & \stackrel{(1)}{\iff} & \frac {EA}{EC}=\frac {DA}{DC} & \iff & m\left(\widehat {EDA}\right)=m\left(\widehat{EDC}\right)\\\\ \frac {CD}{BC}\cdot \frac{PA}{PD}=\frac {DA}{DB} & \stackrel{(2)}{\iff} & \frac {FA}{FB}=\frac {DA}{DB} & \iff & m\left(\widehat {FDA}\right)=m\left(\widehat{FDB}\right)\end{array}\right|$ $\implies$ $DE\perp DF$ .

Reciproc. Suppose $DE\perp DF$ . Denote $L\in DF\cap AE$ . Observe that $(A,C;E,L)$ is an harmonical division. Thus,$DE\perp DF\iff\widehat{EDC}\equiv\widehat{EDA}$ . Thus, $\left\{\begin{array}{c} \frac {EA}{EC}=\frac {DA}{DC}\\\\ \frac {FA}{FB}=\frac {DA}{DB}\end{array}\right|\ (1)$

In conclusion, $\frac {AD}{PD}=1+\frac {PA}{PD}\stackrel{\mathrm{(Van\ Aubel)}}{\implies}\ \frac {AD}{PD}=$ $1+\frac {EA}{EC}+\frac {FA}{FB}\stackrel{(1)}{=}$ $1+\frac {DA}{DC}+\frac {DA}{DB}\implies$ $\frac {AD}{PD}=1+\frac {DA}{DC}+\frac {DA}{DB}\implies$ $\frac{1}{PD}=\frac{1}{AD}+\frac{1}{BD}+\frac{1}{CD}$ .


PP12. Let $ABCD$ be a square and $P\in (CD)$ . Denote $\left\{\begin{array}{c} Q\in BP\ ,\ AQ\perp BP\\\ R\in AP\ ,\ BR\perp AP\\\ S\in CQ\cap DR\end{array}\right|$ . Prove that $SA\perp SB$ .

Proof. Denote $\left\{\begin{array}{c} E\in AQ\cap BC\\\ F\in BR\cap AD\end{array}\right|$ . Since $\widehat{BAE}\equiv\widehat{CBP}$ and $AB=BC$ then $\triangle ABE $ $\stackrel{(\mathrm{s.a.s})}{\cong}$ $ \triangle BCP$ . Thus, $BE=CP$ $\Longrightarrow$ $CE=DP$ .

Likewise, $\triangle BAF \cong $ $\triangle ADP$ $\implies$ $DF=CP$ $\Longrightarrow$ $\triangle PCE$ $\stackrel{(\mathrm{s.a.s})}{\cong}$ $\triangle FDP$ $\Longrightarrow$ $\widehat{EPC}\equiv\widehat{PFD}$ . Since $EQPC$ and $FRPD$ are both cyclically,

due to their right angles at $C,Q$ and $D,R,$ then $\widehat{PRD}\equiv$ $\widehat{PFD}\equiv$ $\widehat{EPC}\equiv$ $\widehat{EQC}$ $\Longrightarrow$ $ARQS$ is cyclically $\Longrightarrow$ $S$ is on the circle with diameter $[AB]$ . Date : 2/5/2013.


PP13. Let acute $\triangle ABC$ with circumcircle $w$ and altitudes $BE$ , $CF$ , where $\left\{\begin {array}{ccc} E\in AC & ; & F\in AB\\\\ M\in BC & ; & MB=MC\end{array}\right\|$ . The circumcircle of $\triangle AEF$ cut again $w$ at $P$ . Prove that $PA\perp PM$

Proof. The pentagon $AEHFP$ is inscribed in the circle with diameter $[AH]$ , where $H\in BE\cap CF$ is orthocenter of $\triangle ABC$ . Let $\{A,S\}=\{A,O\}\cap w$ .

Therefore, $PA\perp PH\implies$ $S\in PH$ . Is well-known or prove easily that $S\in HM\cap AO$ . In conclusion, $M\in PH\implies PA\perp PM$ .


PP14. Let an $A$-right $\triangle ABC$ , $D\in (AC)$ so that $m\left(\widehat{ABD}\right)=C$ and the projection $E$ of $D$ on $BC$ . Prove that $AE=c$ .

Proof. $\triangle ADB\sim\triangle ABC\implies$ $\frac {AD}{c}=\frac {DB}{a}=\frac cb$ $\implies \left\|\begin{array}{c} AD=\frac {c^2}{b}\\\\ BD=\frac {ac}{b}\end{array}\right\|\implies$ $CD=AC-AD=b-\frac {c^2}{b}\implies CD=\frac {b^2-c^2}{b}$ . Also $\triangle CDE\sim\triangle CBA\implies$

$\frac {CD}{a}=\frac {DE}{c}=\frac {CE}{b}\implies$ $\left\|\begin{array}{c} DE=\frac ca\cdot CD\\\\ CE=\frac ba\cdot CD\end{array}\right\|\implies$ $\left\|\begin{array}{ccc} DE=\frac ca\cdot \frac {b^2-c^2}{b} & \implies & DE=\frac {c\left(b^2-c^2\right)}{ab}\\\\ CE=\frac ba\cdot \frac {b^2-c^2}{b} & \implies & CE=\frac {b^2-c^2}{a}\end{array}\right\|\implies$ $BE=BC-EC=a-\frac {b^2-c^2}{a}\implies BE=\frac {2c^2}{a}$ .

Apply the Ptolemy's theorem to the cyclical quadrilateral $ABED\ :$

$AB\cdot DE+AD\cdot BE=BD\cdot AE\iff$ $c\cdot \frac {c\left(b^2-c^2\right)}{ab}$ $+\frac {c^2}{b}\cdot\frac {2c^2}{a}=\frac {ac}{b}\cdot AE\iff$ $c^2\left(b^2-c^2\right)+2c^4=a^2c\cdot AE\iff$ $\boxed{AE=c}$ .

Remark. $m(\widehat{ABD})=C\implies\Delta ADB\sim\Delta ABC\iff m(\widehat{ADB})=B\ .$ Also, $ABED$ cyclic $\implies m(\widehat{AEB})=$ $m(\widehat{ADB})=B\implies AE=AB$ .


PP15. Let $\triangle ABC$ with $A > 90^\circ$ , circumcenter $O$ and orthcenter $H$ of the triangle $ABC$ . Define the reflection $K$ of $H$ w.r.t. $A$ . Prove that $K\in OC\iff A=90^{\circ}+B$ .

Proof 1. Let the midpoint $M$ of $[BC]$ and diameter $[CD]$ of circumcircle $(O)$ . Thus, $DB=2\cdot OM=AH\implies$ $\boxed{BD=AK}\ (*)$ and $m\left(\widehat{BCD}\right)=A-90^{\circ}$ and

$A=90^{\circ}+B\iff$ $m\left(\widehat{BAD}\right)=B\iff$ $AB\parallel CD$ , i.e. $ABDC$ is isosceles trapezoid $\stackrel{(*)}{\iff}$ $ABDK$ is parallelogram $\iff K\in CD\iff K\in OC$ .

Proof 2. See $\triangle ACK$ , where $AK=-2R\cos A$ . Thus, $K\in OC\iff$ $m\left(\widehat {ACK}\right)=A+C-90^{\circ}$ and $m\left(\widehat {AKC}\right)=180^{\circ}-A\iff$ $\frac {AK}{\sin\widehat {ACK}}=\frac {AC}{\sin \widehat{AKC}}\iff$

$\frac {-2R\cos A}{\sin \left(A+C-90^{\circ}\right)}=\frac {2R\sin B}{\sin\left(180^{\circ}-A\right)}\iff$ $\frac {\cos A}{\cos (A+C)}=\frac {\sin B}{\sin A}\iff$ $\sin A\cos A=-\sin B\cos B\iff$ $\sin 2A+\sin 2B=0\iff$

$2\sin (A+B)\cos (A-B)=0\iff$ $\sin C\cos (A-B)=0\iff$ $\cos (A-B)=0\iff$ $|A-B|=0\iff$ $A=B+90^{\circ}$ .


PP16. Let $ABC$ be a triangle with the incircle $int=C(I,r)$ and the circumcircle $ext=C(O,R)$ .

Let $D\in BC\cap int$ , i.e. $ID=r$ , $\{A,S\}=AI\cap ext$, $\{S,P\}=\{S,D\}\cap ext$ . Prove that $PI\perp PA$ .

Proof 1. Denote the diameter $[AA']$ of $ext$ , i.e. $AA'=2R$ . Is well-known that $\boxed{\widehat{DIS}\equiv\widehat{IAA'}}\ (1)$ , $SB=SC=SI$ and $\boxed{IA\cdot IS=2Rr}\ (2)$ from the power

of $I$ w.r.t. $ext$ . Therefore, $\left\{\begin{array}{cc} (1) & \widehat{DIS}\equiv\widehat{IAA'}\\\\ (2) & \frac {DI}{IA}=\frac {IS}{AA'}\end{array}\right\|$ $\stackrel{s.a.s}{\iff}$ $\triangle DIS\sim\triangle IAA' $ . In conclusion, $\widehat{ISD}\equiv\widehat{AA'I}\iff$ $P\in SD\cap A'I\iff$ $PI\perp PA$ .

Proof 2. I"ll use the notations from the previous method. Thus, $\widehat{DBS}\equiv \widehat{BPS}\iff$ $\triangle SBD\sim\triangle SPB\iff$ $SD\cdot SP=SB^2$ $\iff$

$SD\cdot SP=SI^2\iff$ $\triangle SDI\sim\triangle SIP\iff$ $\widehat {SPI}\equiv\widehat{SID}\iff$ $\widehat{SPI}\equiv\widehat{SAA'}\iff$ $A'\in PI\iff$ $PA\perp PI$ .

Proof 3. Denote $E\in BI\cap AC$ and $F\in CI\cap AB$ . Thus, $m\left(\widehat{BPS}\right)=m\left(\widehat{CPS}\right)= \frac A2\implies$ $\frac{{DB}}{{DC}} = \frac{{PB}}{{PC}}$. Since $\frac{{BF}}{{CE}}=\frac{{BD}}{{CD}}$ obtain that $\frac{{PB}}{{PC}}=\frac{{BF}}{{CE}}$ . Since

$\widehat{FBP}\equiv\widehat{ECP}$ obtain that $\triangle FBP\sim\triangle ECP\implies$ $\widehat{AFP}\equiv\widehat{AEP}\implies$ $E,F,P$ belong to the circle with the diameter $[AI]\implies$ $PI\perp PA$ . Thank you, Tsikaloudakis !


PP17 (Bulgarian IMO TST 2005). Let acute $\triangle ABC$ with circumcircle $w=C(O,R)$ , incenter $I$ , orthocenter $H$ and $AB<AC$ . The lines $AH$

and $AI$ meet again $w$ at the points $D$ and $L$ respectively. Prove that $IH\perp IA\iff \cos B+\cos C=1\iff$ $r=R\cos A\iff$ $DI\perp DL$ .

Proof 1 (synthetic). Denote $\{L,N\}=LO\cap w$ , $P\in AD\cap BC$ and $S\in AI\cap BC$ . Therefore:

$\blacktriangleright\ DI\perp DL\iff$ $m\left(\widehat{DIL}\right)=B-C\iff$ $m\left(\widehat{ADI}\right)=\frac {B-C}{2}\iff$ $N\in DI\iff$ $ADLN$ is an isosceles trapezoid $\iff$ $IO\parallel BC\iff$ $r=R\cos A$ .

$\blacktriangleright\ IH\perp IA\iff$ $HPSI$ is cyclically $\iff AI\cdot AS=AH\cdot AP\iff$ $\frac {b+c}{2s}\cdot\frac {4bcs(s-a)}{(b+c)^2}=2Rh_a\cos A\iff$ $\frac {bc(b+c-a)}{b+c}=bc\cos A\iff$

$b+c-a=(b+c)\cos A\iff$ $(b+c)(1-\cos A)=a\iff$ $2\sin^2\frac A2(\sin B+\sin C)=\sin A\iff$ $\cos B+\cos C=1$ .

Proof 2 (trigonometric).

$\blacktriangleright\ IH\perp IA\iff$ $AI=AH\cos \frac {B-C}{2}\iff$ $\frac {s-a}{\cos\frac A2}=2R\cos A\cos\frac {B-C}{2}\iff$ $s-a=R\cos A(\sin B+\sin C)\iff$

$2(s-a)=(b+c)\cos A\iff$ $(b+c)(1-\cos A)=a\iff$ $2\sin^2\frac A2(\sin B+\sin C)=\sin A\iff$ $\cos B+\cos C=1$ .

$\blacktriangleright\ DI\perp DL\iff$ $m\left(\widehat{DIL}\right)=B-C\iff$ $DL=IL\sin(B-C)\iff$ $2R\sin\frac {B-C}{2}=IL\sin (B-C)\iff$ $R=IL\cos\frac {B-C}{2}\iff$

$OI\parallel BC\iff$ $r=R\cos A\iff$ $\iff$ $AD=2\cdot IA\cos \frac A2\iff$ $\cos A=\frac rR\iff$ $\cos A=\cos A+\cos B+\cos C-1\iff$ $\cos B+\cos C=1$ .


PP18. Let a rectangle $ ABCD$ with center $ O$ and $ AB\neq BC$. Let $ E\in AB$ , $ F\in BC$ so that $ O\in EF$

and $ EF\perp BD$ . Let $ M,N$ be the midpoints of $ [CD]$ , $ [AD]$ respectively. Prove that $ FM \perp EN$.

Proof. Denote $ L\in FM\cap EN$ . Observe that the quadrilaterals $ OAED$ , $ OFCD$ are cyclically (diameters $ [DE]$ . $ [DF]$ respectively). Thus, $ \left\|\begin{array}{c} \widehat {DEF}\equiv\widehat {DEO}\equiv\widehat {DAO}\equiv\widehat {DAC} \\ \\ \widehat {DFE}\equiv\widehat {DFO}\equiv\widehat {DCO}\equiv\widehat {DCA}\end{array}\right\|$

$ \implies$ $ \boxed {\ \triangle DEF\ \sim\triangle DAC\ }$ $ \implies$ $ DE\perp DF\ \implies\ BEDF$ is cyclically (diameter $ [EF]$ ) $ \implies$ $ \widehat {CFD}\equiv\widehat {AED}$ $ \implies$ $ \boxed {\ \triangle CFD\ \sim\ \triangle AED\ }$ $ \implies$ $ \widehat {CMF}\equiv\widehat {ANE}$ $ \implies$

$ \widehat {CML}\equiv\widehat {DNL}$ $ \implies$ the quadrilateral $ DMLN$ is cyclically with the diameter $ [MN]$ .Therefore, $ LM\perp LN$ $ \implies$ $ FM\perp EN$ . Otherwise, if denote the intersection $ P\in AD\cap EF$ ,

then $ BP\parallel FD$ and $ P$ is the orthocenter of the triangle $ BDE$ , i.e. $ BP\perp DE$ . Therefore, $ DE\perp DF$ a.s.o.

Remark II. Generally, for $ N\in (AD)$ , $ M\in (CD)$ so that $ \frac {NA}{ND} = \frac {MC}{MD}$ $ \implies$ $ FM\perp EN$ . $ \triangle DAE$ results from $ \triangle DCF$ by rotation $ \left(D\ ,\ 90^{\circ}\right)$ and homothety $ \left(D\ ,\ \frac {DA}{DC}\right)\ !$


PP19. Let a square $ABCD$ and $\left\{\begin{array}{c} X\in (AD)\\\\ M\in (AB)\\\\ N\in (BC)\end{array}\right\|$ so that $AX=BM=BN$ . Denote $Y\in MC\cap NX$ , $L\in MC\cap BX$ and $Z\in LN\cap CD$ . Prove that $DY\perp BZ$ .

Proof 1 (analytic). Let $\left\{\begin{array}{ccc} A(0,0) & B(0,1) & C(1,1)\\\\ D(1,0) & X(m,0) & N(m,1)\\\\ Y(m,y) & M(0,1-m) & Z(1,1+z)\end{array}\right\|$ . From $Y\in MC$ obtain that $\left|\begin{array}{ccc} 0 & 1-m & 1\\\\1 & 1 & 1\\\\ m & y & 1\end{array}\right|=0\iff$ $y=m^2-m+1\implies$ the slope of $DY$

is given by the relation $\boxed{s_{DY}=-\frac {m^2-m+1}{1-m}}$ .Thus, $L\in XB\cap MC \implies$ $\left\{\begin{array}{cc} XB\ : & y=\frac {-(x-m)}m\\\\ MC\ : & y-1=m(x-1)\end{array}\right\|\implies$ $-\frac xm=m(x-1)\ \wedge\ y=1-\frac xm\implies$

$L\left(\frac {m^2}{m^2+1},\frac {m^2-m+1}{m^2+1}\right)$ . Observe that $\boxed{s_{BZ}=z}$ , where $Z\in LN\cap CD$ , i.e. $\left|\begin{array}{ccc} 1 & 1+z & 1\\\\ m & 1 & 1\\\\ m^2 & m^2-m+1 & m^2+1\end{array}\right|=0$ $\iff$ $\left|\begin{array}{ccc} 1 & z & 0\\\\ m & 1-m & 1-m\\\\ m^2 & 1-m & 1\end{array}\right|=0$ $\iff$

$(1-m)+m^2(1-m)z=(1-m)^2+mz\iff$ $\boxed{z=\frac {1-m}{m^2-m+1}}$ . Therefore, $s_{BZ}\cdot s_{DY}=-1$ , i.e. $DY\perp BZ$ .

Proof 2 (synthetic). Let $T\in CD\cap BL$ . Thus, $\frac {CT}{BM}=\frac {LC}{LM}=\left(\frac {BC}{BM}\right)^2\implies$ $\boxed{BC^2=BM\cdot CT}\ (*)$ . Otherwise, $BCM\sim CTB\iff \frac {BC}{CT}=\frac {BM}{CB}$ , i.e. $(*)$ . So

$\frac {XD}{YX}=\frac {NC}{NY}\cdot\frac {NY}{YX}=$ $\frac {BC}{BM}\cdot \frac {ZC}{CT}\ \stackrel{(*)}{=}\ \frac {BC}{BM}$ $\cdot\frac {ZC}{\frac {BC^2}{BM}}=$ $\frac {ZC}{BC}\implies$ $\frac {XD}{XY}=\frac {ZC}{BC}\implies$ $\triangle DXY\sim\triangle ZCB\implies$ $\widehat {DYX}\equiv\widehat {ZBC}\implies$ $DY\perp ZB$ because $XY\perp CB$ .

PP20. Let $\triangle PAB\ ,\ \triangle QAC$ outside of $\triangle ABC$ such that $\left\{\begin{array}{c} AP = AB\\\\ AQ = AC\\\\ \widehat{BAP}\equiv\widehat{CAQ}\end{array}\right\|\ .$ Let $R\in BQ\cap CP$ and the circumcenter $O$ of $\triangle BCR\ .$ Prove that $AO \perp PQ\ .$
Proof. I"ll show that $OP^{2}-OQ^{2}=AP^{2}-AQ^{2}$, i.e. $OA\perp PQ\ .$ Let $2x=m(\widehat{BAP})\ ,$ the circumcircle $C(O,\rho )$ of $\triangle BRC$ and $S\in AR\cap BC\ .$ Thus, $\left\{\begin{array}{c}AC=AQ\\\\ AP=AB\\\\ \widehat{CAP}\equiv\widehat{QAB}\end{array}\right\|$
$\implies$ $\triangle ACP\sim\triangle AQB\ .$ Thus, $\left\{\begin{array}{c}CP=QB\\\\ \widehat{APC}\equiv\widehat{ABQ}\\\\ \widehat{ACP}\equiv\widehat{AQB}\end{array}\right\|$ $\implies$ $\left\{\begin{array}{c}\widehat{APR}\equiv\widehat{ABR}\\\\ \widehat{ACR}\equiv\widehat{AQR}\end{array}\right\|$ $\implies$ the quadrilaterals $APBR\ ,$ $AQCR$ are cyclically with the radical axis $AR\ .$ Thus,

$\left\{\begin{array}{c}m(\widehat{BRS})=m(\widehat{BPA})=90^{\circ}-x\\\\ m(\widehat{CRS})=m(\widehat{CQA})=90^{\circ}-x\\\\ m(\widehat{BRP})=m(\widehat{BAP})=2x\end{array}\right\|$ $\implies$ the ray $[RS$ is bisector of $\widehat{BRC}\ .$ Prove easily that $\left\{\begin{array}{c} m(\widehat{BOC})=4x\\\\ a=2\rho\sin 2x\end{array}\right\|\ .$ Thus, $\left\{\begin{array}{c} PA=c\ ,\ QA=b\\\\ PB=2c\cdot\sin x\\\\ QC=2b\cdot \sin x\end{array}\right\|$ and

$\left\{\begin{array}{c} m(\widehat{CBO})=m(\widehat{BCO})=90^{\circ}-2x\\\\ m(\widehat{PBO})=180^{\circ}+3x-B\\\\ m(\widehat{QCO})=180^{\circ}+3x-C\end{array}\right\|\ .$ Apply generalized Pythagoras' theorem in $\triangle POB\ ,$ $\triangle QOC$ respectively $:\ \left\{\begin{array}{c} OP^{2}=BP^{2}+\rho^{2}-2\rho\cdot BP\cos \widehat{PBO}\\\\ OQ^{2}=CQ^{2}+\rho^{2}-2\rho\cdot CQ\cos \widehat{QCO}\end{array}\right\|$

$\implies$ $OP^{2}-OQ^{2}=4(c^{2}-b^{2})\sin^{2}x+4c\rho\sin x\cos (B-3x)-4b\rho \sin x\cos (C-3x)=$ $4(c^{2}-b^{2})\sin^{2}x+$ $4c\rho \sin x(\cos B\cos 3x+\sin B\sin 3x)$

$-4b\rho \sin x(\cos C\cos 3x+\sin C\sin 3x)=$ $4(c^{2}-b^{2})\sin^{2}x+$ $4\rho\sin x\cos 3x(c\cos B-$ $b\cos C)+$ $4\rho \sin x\sin 3x(c\sin B-b\sin C)=$

$4(c^{2}-b^{2})\sin^{2}x+4\rho\sin x\cos 3x\left(\frac{a^{2}+c^{2}-b^{2}}{2a}-\frac{a^{2}+b^{2}-c^{2}}{2a}\right)=$ $4(c^{2}-b^{2})\sin^{2}x+\frac{4\rho (c^{2}-b^{2})}{a}\sin x\cos 3x=$

$\left(c^{2}-b^{2}\right)\left(4\sin^{2}x+\frac{2\sin x\cos 3x}{\sin 2x}\right)=c^{2}-b^{2}\ .$ I used the simple relations $c\sin B=b\sin C$ and $4\sin^{2}x+\frac{2\sin x\cos 3x}{\sin 2x}=1\ .$


PP21 (F.J.G. Capitan). Let $\triangle ABC$ with incircle $w=C(I,r)$ which touches it at $\left\{\begin{array}{c} D\in BC\\\\ E\in CA\\\\ F\in AB\end{array}\right\|$ . For a mobile $M\in AI$ denote $\left\{\begin{array}{c} X\in MB\cap DF\\\\ Y\in MC\cap DE\end{array}\right\|$ . Prove that $XY\perp AI$ .

Proof. Let $\delta_d(P)$ - the distance from $P$ to $d$ . Thus, $\frac {XF}{XD}=\frac {\delta_{AB}(X)}{\delta_{BC}(X)}=$ $\frac {\delta_{AB}(M)}{\delta_{BC}(M)}\ \stackrel{(M\in AI)}{=}\ \frac {\delta_{AC}(M)}{\delta_{BC}(M)}= $ $\frac {\delta_{AC}(Y)}{\delta_{BC}(Y)}=$ $\frac {YE}{YD}\implies$ $\frac {XF}{XD}=\frac {YE}{YD}\implies$ $XY\parallel EF\implies$ $XY\perp AI$ .
This post has been edited 132 times. Last edited by Virgil Nicula, Jul 23, 2016, 10:47 PM

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+ September 2012
357. Morley theorem.
356. Order relation between s & (2R+r) in an acute triangle.
+ August 2012
355. Some problems with polynomials or equations I
354. Gradul II - Univ. B.B. Cluj, aug. 1998 Prof I sau II.
353. Some problems of the analytical geometry.
+ July 2012
352. Some definite integrals.
351. Problems with areas.
350. Another (easy) integrals.
349. Another interesting limits.
348. Subiectele de la BAC 2012 /Mate-Info.
+ June 2012
347. Two isogonal lines in a triangle.
346. Some easy integrals.
345. Characteristic function of a set.
344. Some trigonometry problems.
343. Problems with collinear points and concurrent lines.
+ May 2012
342. Exponential or logarithmic equations/inequations.
+ April 2012
341. Some nice and interesting "slicing" problems.
340. Geometry problems for middle school.
339. Derivatives. Rolle's function.
338. ROU National Math Olympiad 2012.
+ February 2012
337. Nice metrical problems.
+ January 2012
336. Some problems of the analytical geometry.
335. Some difficult geometric/trigonometric inequalities.
334. 2011 Japan Mathematical Olympiad Finals Problem 1.
333. Problems with the common tangent for two circles.
332. Some metrical problems from the contests I.
+ December 2011
331. Some nice problems with polynomials/equations I.
+ November 2011
330. A triangle and an its transversal through a point.
329. Some problems with complex numbers.
328. A general identities with areas in a triangle ABC.
+ October 2011
327. Some geometrical/algebraic extremum problems.
326. Some problems from trigonometry.
325. Without the l'Hospital's rules.
324. Some determinants.
323. Easy, nice and interesting problems for high school.
322. Incenter, Gergonne, Nagel a.s.o. of ABC.
321. Harmonical quadrilateral.
320. Some problems with metrical relations.
+ September 2011
319. Some problems with induction.
318. IRAN - 2011 (geometrie).
317. Some interesting geometry problems for middle school.
316. Some properties of isogonal rays in an angle of ABC.
+ August 2011
315. Indian Team Selection Test 2010 ST1 P1 and extension.
314. USAMTS 2009 Round 2 #5.
313. For middle school - some nice problems.
312. Very nice (famous trigonometrical problems) !
311. Some problems from USAMO and other contests.
310. Some properties of the tangential quadrilateral.
309. Beautiful problem ! IRAN Selection Test for IMO, 2004.
308. Some nice and easy properties in a triangle.
+ July 2011
307. Very nice proofs !
306. Some easy and nice "slicing" problems.
305. Position of the roots w.r.t. a given real number.
304. Integrals ("brothers") III.
303. Integrals II.
302. Some problems with sequencies of the "roots" .
301. Some trigonometrico-geometrical inequalities.
300. Lagrange's multiplier. Examples and problems.
299. A class of the recurrent sequencies.
298. Some nice problems from Crux Mathematicorum.
297. Nice inequalities with concrete numbers.
296. Some nice Vittasko's problems.
295. M1 3th subject, 2c - School Graduate, ROMANIA.
294. The Octav Halunga's inequality in a triangle.
+ June 2011
293. A difficult trigonometric equation (third point !).
292. Length of the angled bisector and some means.
291. Some integrals.
290. Characterize "OP perpendicular on OA" in ABC a.s.o.
289. Nice identities in an algebra/ geometry/trigonometry.
288. An inequality with two A-cevians in a triangle ABC.
287. Some metrical problems with or without trigonometry.
286. ABC is right triangle iff s=2R+r and other similar pp.
285. Nice and easy inequalities. For example, IMAC - 2009.
284. Ellipse (definition).
283. Some nice and easy problems.
282. Factorize of polynomials.
+ May 2011
281. Identity with R in an acute triangle (ILL - 1974).
280. Common roots of two polynomial equations.
279. O.M. Holland , O.M. Ireland 1988 and TST USA 2000, etc.
278. In memoria profesorului Alexandru Lupas.
277. A nice and difficult relation with Lemoine's axis.
276. Chinese TST 2009 and Second Round 2006.
275. Some interesting algebraic equations and systems.
274. An inequality in a triangle for its interior point.
273. The area of the triangle IOH. When I belongs to OH ?
272. Outside of triangle. Extension of Fermat's point.
271. Relations concerning Fermat points.
+ April 2011
270. Morrocan M.O. 2010 and Croatia TST 2009.
269. Some nice and easy properties in a trapezoid.
268. Saraghin contest, 2011.
267. Germany 2002, Russia 2003, IMO 2003, Aus.-Polish 1992.
266. Rectangles erected on sides of a triangle. Concurrence.
265. Really nice problem !
264. Identities with complex numbers.
263. Newton's sums. Applications.
262. IMO Shortlist 2005, Geometry 6th.
261. IMO 2009, problem 2 and JBMO 2007, problems 1 & 2.
260. Some nice, simple identities and their applications.
259. ABC equilateral triangle ==> DEF equilateral triangle.
258. Some difficult and nice problems with complex numbers.
257. Seeking roots of 2th equation with complex coef.
256. An extremum problems with complex numbers.
+ March 2011
255. A line which is parallelly with OH.
254. An interesting C. Mateescu's geometrical inequality.
253. An old algebraic inequality.
252. OI=f(A,R) in certain conditions.
251. Problems of Analytical Geometry.
250. An application of the Euler's relation.
249. An usual and nice metrical relations in quadrilateral.
248. Pagina instructiva.
247. An remarkable inequality with median.
246. R1-R2=OI (nice).
245. An inequality in an acute triangle.
244. Some properties of cyclic orthodiagonal quadrilaterals.
243. Nice ! (a+b) is constant.
242. A general geometrical problems.
241. Areas in a triangle.
240. Some interesting geometrical inequalities.
239. Algebraic marathon.
238. Some metrical problems.
237. Some problems with ... problems.
236. A locus with the metrical relation.
235. Proofs of some geometry problems with complex numbers.
234. Some simple and nice problems from contests.
233. An interesting identity and an easy & nice inequality.
+ February 2011
232. Some geometrical inequalities (own).
Pagina libera
231. A conditioned minimum value.
230. Some usual synthetical properties in a triangle.
229. Concurs online MATEFBC, ed. 5 -subiect 2.
228. OJM - Romania, 2010 problem 3.
227. IMO ShortList 2003 (problem 5).
226. Some inequalities from Calculus (Real Analysis).
225. An interesting problems from Kunny.
224. Some nice and easy inequalities in a triangle.
223. An inequality with complex numbers-RMO shortlist 2010.
222. An extension of probl. 2 from IMO 2009.
221. Some nice and simple "slicing" problems.
220. Some nice geometry problems from contest.
+ January 2011
219. A very nice geometrical inequality.
218. Some problems from the Suiss IMO Selection Team 2006.
217. A nice and interesting Murray S. Klamkin's problem.
216. Nice ! Find the length of hypotenuse (Belarus - 2010)
215. Limit of the unique root for polynominal equations.
214. A range of some functions.
213. Another classical "slicing" problems.
212. A simple applications of the Casey's theorem.
211. Pretty hard problem !
210. Tangential circle of the triangle ABC.
209. Nice problem, nice proof !
208. An interesting geometrical inequality.
207. Some interesting problems of Toshio Seimiya, Japan.
206. Four problems with extremum in a quadrilateral.
205. A "slicing" problem with 11 methods.
204. Saraghin 2010 (three geometry problems).
203. An application of the Pascal's permutation theorem.
+ December 2010
202. Concurrent lines in a right triangle.
201. Applications of Desarques' and Pappus' theorems.
200. Some nice applications of the inversion.
199. Some properties of symmedian. Two extensions.
198. Generalized Carnot's lemma.
197. Triangles outside of a triangle and concurrence.
196. Similar rectangles outside of a triangle.
195. Semicircle. Nice application of harmonical division.
194. The Lalescu's sequence. Properties.
193. An inequality with n! .
192. An easy and nice problem of Valentin Vornicu.
191. Six nice problems with the incircle of a triangle.
190. Very nice ! Coculescu's Contest seniors 2010.
189. Pascal's theorem and some nice and easy applications.
188. Some problems from vectorial geometry.
187. Distancies.
186. Without the l'Hospital's rules.
185. The Durrande's relation.
184. An interesting geometry problem from RMO 2010.
183. A remarkable characterization of equilateral triangle.
182. Another two nice problems with complex numbers (own).
181. Two special inequalities from CRUX (own).
180. Own proposed problem from CRUX (with complex numbers).
+ November 2010
179. The range of |z| , |z^2+a|=|bz+c|. Particular cases.
178. Some nice and simple geometry problems.
177. A nice proof of one identity in a triangle.
176. Another nice problems for middle school.
175. A Geometric Proof (without trig.) of Heron's Formula.
174. Some interesting properties of modulus (middle school).
173. An interesting trigonometric equations.
172. The Zelinsky's problem.
171. Another "slicing" proposed problems (middle school).
+ October 2010
170. A nice "slicing" proposed problem for middle school.
169. Some metrical problems in a circle (III).
168. Some properties of the Lemoine's point.
167. Nice problems - OIM, 1995 and ... Toshio Seimyia.
166. Some constant geometrical expressions.
165. Some metrical problems (5) in a circle (II).
164. Some metrical problems (5) in a circle.
163. Applications of Ptolemy's inequality. Ex. - China,1998.
162. Generalization of proposed problem from Mongolia, 2000.
161. A difficult inequality (barycentrical coordinates)
160. A very nice "slicing" problem.
159. Multiple derivatives in a point.
158. Very nice and difficult limits.
157. Steinbart's theorem. Applications.
156. Some problems with perpendicularities.
155. A constant ratio in a triangle ABC.
154. Difficult inequality (without derivatives eventually).
153. Geometrical minimum/maximum.
152. Concurrence, cyclically, metrics, symmedian a.s.o.
151. Miscellaneous problems for the admission at math.
150. Some (5) problems with fixed points.
149. Some (8) problems with collinearity or concurrence.
148. A special class of systems.
147. A problem with two polynominals.
146. Synthetic, metric, trigonometric and analitycal proofs.
145. Spain MO P5 - Point on median.
144. A conditioned concurrency in a triangle.
143. Assess of a ratio (nice and difficult problem).
142. Inscribed/circumscribed quadrilateral w.r.t. ABC.
141. N is interior <==> 1/3<a/b<3 (CRUX Mathematicorum).
140. A synthetical property <==> linear/angled relation.
139. A caracterization of a right angle in a triangle.
138. Generalized "slicing" problems (own).
137. Stewart's relation.
136. A geometrical interpretation of a system (own).
135. Three strong geometrical inequalities.
+ September 2010
134. Some problems with projections/perpendiculars (2).
133. Some problems with projections/perpendiculars (1).
132. Asupra unei inegalitati intr-un triunghi.
131. A stronger Form of the Steiner-Lehmus Theorem (own).
130. A triangle and a fixed point.
129. BX=CY and three properties.
128. Value of IG and two inequalities in neighbourhood of 0.
127. Pagina educativa.
126. Routh's theorem.
125. I like very much this Darij Grinberg's message.
124. A nice and easy problem with a right-angled triangle.
123. My extension and its proof.
122. An identity for an equilateral triangle. Extension.
121. Viete's relations with cos A , cos B , cos C.
120. Two equal neighbouring angles/sides in a quadrilateral.
119. Inspired by a Miculita's problem.
118. An usual remarkable geometrical locus.
117. Some applications of harmonic division/quadrilateral.
116. The length of AE (nine methods !).
115. An difficult equation with one radical.
114. A property of tangential triangle for ABC.
113. Range for a function.
112. The equivalence relation ".s.s."
111. Simple, but interesting ...
110. German TST 2004 - Exam V , Problem 1
109. Two equal areas ==> find A.
108. Old, short and nice proof of Euler's relation.
107. XY parallel BC and X,Y are isogonal/izotomic points.
106. OLM - Bucuresti (geometrie).
105. A problem with three circles.
104. Concursul Revistei de Matematica din Galati (RMG).
103. Metrical usual relations in triangle. Applications.
102. Characterization of a metrical relation in a triangle.
101. O problema cu numere complexe.
100. Bobiller's theorem.
99. Some simple and nice geometry problems.
98. Symmedian & median (metrical relations).
97. Problem of butterfly.
+ August 2010
96. Integrals.
95. Three circles in an angle.
94. Equations of angle bisectors (interior and exterior).
93. The Appolonius' construct problems.
92. CA , CB tangent to parabola - OIMU 2008 Problem 4.
91. JBMO 2010, Problem 3.
90. Nice and difficult inequality in ABC (own).
89. Similarity (parallel/antiparallel).
88. A nice geometric locus conditioned metrically.
87. An interesting vectorial identity.
86. A very nice maxmin.
85. Problem "slicing" : 60-24-12.
84. IRAN national math olympiad - 2010
83. A metrical characterization of concurrence.
82. AA' , BN , CM concur in Gergonne point.
81. Sawayama and Thebault’s theorem.
80. Three concurrent perpendiculars (the Carnot's lemma).
79. A quadrilateral with many perpendicularities.
78. Jakobi's theorem.
77. Two equivalent perpendicularities.
76. A parallelogram and a circle (nice !).
75. Extended Steinbart Theorem.
74. Inegalitatea Erdos-Mordell.
73. A nice problems with a square and a perpendicularity.
+ July 2010
72. Colinearity in a triangle - H , P , M .
70. Colinearity with pole/polar - H\in PQ .
71. An angle questions (synthetic/trigonometric proofs).
69*1. Slicing problem 3.
69. Position of x1, x2 w.r.t. a given real number k.
68. Some nice applications of "pole/polar".
67. Remarkable algebraic/geometrical inequalities (1).
66. Remarkable perpendicularities in a triangle.
65. Parallel to BC through I .
64. Problems of the type "instant" (at most one minute).
63. A nice problem with radical center (livetolove212).
62. Nice and easy problem with a right triangle.
61. IMO Shortlist 2009 (very nice problem and its proof).
60. Mediteranean M.C. 2010 Problem 3.
59. IMO 2010, problem 4.
58. A cevian and two incircles.
57. Opinii si comentarii relativ la inv. rom. II
56. A extremum problem with areas.
55. Two important circles - mixtilinear incircle/excircle.
54. Cinci grade de libertate.
53. An remarkable concurrency.
52. Harmonical division.
51*1. JBTST-3/2010 Aplicatii la proprietatea fluturelui.
51. A nice problem with perpendicularity from Jaime.
50. IMAC 2010 seniors, the first day.
49. The midpoint of a cevian in an acute triangle.
+ June 2010
48. An inequality in a triangle : h_a+max{b,c} ...
47. Interesting and difficult identities in a triangle.
46. Outside of a quadrilateral.
45. \cos B+\cos C=1 <==> nice property.
44. Relatia IA=IO sau IA=IM intr-un triunghi ABC etc.
43. A "slicing" problem with a parameter.
42. IMAC 2010 Problema 2.
41. ONM 2010 Calarasi Problema 3.
+ May 2010
40. Lema lui Haruki.
39. A nice and remarkable problem with Brocard's point.
38. Minimum area of triangle touching with semicircle.
37. Own extension of the Steiner-Lehmus' problem.
36. ONM (juniori) - 2010 Iasi, problema 4.
35. Nice problem from Arqady (Eduard_Tur).
34. IMO Shortlist 2002 geometry problem 7.
+ April 2010
32. Linkuri diverse.
31.Some nice metrical problems.
29. Inegalitate integrala de la Kunny.
28. A fost odată ...
27.Problem of Darij Grinberg (I - centroid of triangle DEF).
26. Outside of a triangle.
25. Geometric equations.
24. A fine and cool inequalities.
23. A very nice exponential inequality (own - from CRUX).
22. Some interesting limits (sequence/function).
21 bis. Some problems with complex numbers.
21. Probleme de geometrie (Yetty & I).
20. O ineg.cu suma de AG/AI .
19. Own proposed problems in CRUX Mathematicorum - Canada.
18. O problema a lui Toshio Seimiya.
17. O inegalitate "tare" intr-un triunghi.
16. Length of the (interior) angle bisector (10 proofs).
15. Inequalities stronger than Panaitopol's.
14. Opinii si comentarii relativ la inv. rom.
13. Inegalitatea de la Sorin Borodi.
12. Inegalitatea I. Maftei & S. Radulescu (2).
11. Inegalitatea I. Maftei & S. Radulescu (1).
10. Walker's inequality and its extension.
9. Inegalitatea HO+HA+HB+HC >= 3R .
8. The first problem Shortlist ONM 2010 Romania.
7. Some interesting "slicing" problems.
6. Interesting problems from JBMO.
5. Theoretical preliminary for inequalities.
4. Remarkable geometrical inequalities (2).
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