134. Some problems with projections/perpendiculars (2).

by Virgil Nicula, Sep 30, 2010, 8:57 AM

Quote:
PP1.1. (Iran NMO, 2008-6). Let $ABC$ be an acute triangle with the circumcircle $w=C(O,R)$ . Denote the midpoints $E$ , $F$ of $[AC]$ , $[AB]$ respectively,

the projection $D$ of $A$ on $BC$ and the projections $X$ , $Y$ of $D$ on $AB$ , $AC$ respectively. Prove that $OX=AE\iff OY=AF\iff h_a=R$ .


PP1.2. (Iran NMO, 2008-2). The $A$-exincircle $w=C(I_a)$ of $\triangle ABC$ touches $AB$ , $AC$ at $B'$ , $C'$ respectively. Denote the incenter $I$ ,

$P\in I_aB\cap B'C'$ , $Q\in I_aC\cap B'C'$ , $M\in BQ\cap CP$ . Prove that the quadrilateral $IBMC$ is a parallelogram.


PP2. Let $ABC$ be a triangle. For a point $M\in AC$ define the point $N\in BC$ so that $MN\perp BC$ , the point $P$ so that

$PB\perp BC$ , $PC\perp AN$ and the intersection $Q\in PM\cap AN$ . Prove that $AP\perp CQ\iff AB\perp AC$ .


PP3. Let $ABC$ be a triangle with the orthocenter $H$ and the circumcircle $w=C(O,R)$ . Denote $D\in BC$

so that $AD\perp BC$ , the symmetrical point $E$ of $C$ w.r.t. $D$ and $F\in EH\cap AB$ . Prove that $DF\perp DO$ .


PP4. Let $d_1$ , $d_2$ be two lines, where $d_1\parallel d_2\ \ \vee\ \ d_1\cap d_2\ne \emptyset$ . For a point $I$ denote $X\in d_1$ , $Y\in d_2$

so that $IX\perp d_1$ , $IY\perp d_2$ . Suppose that $IX=IY$ . Consider two points $M\in d_1$ , $N\in d_2$ .

Prove that $\boxed{\ IN\perp MY\ \iff\ IM\perp NX\ \iff\ MX^2+NY^2=MN^2\ }$ .

Particular cases

$\blacktriangleright\ d_1\parallel d_2$ . Let $XYNM$ be a trapezoid, where $d_1\equiv XM\parallel YN\equiv d_2$ . Denote the midpoint $I$

of $[XY]$ . Prove that $IN\perp MY\ \iff\ IM\perp NX\ \iff\ MX^2+NY^2=MN^2$ .

$\blacktriangleright\ d_1\cap d_2\ne\emptyset\ \wedge\ XY$ doesn't separates $M$ , $N$ . Let $ABC$ be a triangle with the incircle $w=C(I,r)$ which touches

the sides at $D\in BC$ , $E\in CA$ , $F\in AB$ . Denote $L\in BC\cap EF$ . Prove that $LI\perp AD$ and $LA^2=LD^2+AE^2$ .

Let $ABC$ be a triangle with the exincircle $w_a=C(I_a,r_a)$ which touches the sides at $D_a\in BC$ , $E_a\in CA$ ,

$F_a\in AB$ . Denote $L_a\in BC\cap E_aF_a$ . Prove that $L_aI_a\perp AD_a$ and $L_aA^2=L_aD_a^2+AE_a^2$ .

$\blacktriangleright\ d_1\cap d_2\ne\emptyset\ \wedge\ XY$ separates $M$ , $N$ .

Proof of PP1.1. Observe that $\left\|\begin{array}{ccccc} DX^2=XA\cdot XB & ; & DX=h_a\cos B & ; & OE=R\cos B\\\ DY^2=YA\cdot YC & ; & DY=h_a\cos C & ; & OF=R\cos C\end{array}\right\|$ .

Therefore, $\left\|\begin{array}{cccc} XA\cdot XB=R^2-OX^2 & \implies & OX^2=R^2-DX^2\\\ YA\cdot YC=R^2-OY^2 & \implies & OY^2=R^2-DY^2\end{array}\right\|$ . Thus,

$OX=AE \iff R^2-AE^2=DX^2 $ $\iff OE=DX \iff h_a=R\ .$

$OY=AF \iff R^2-AF^2=DY^2 $ $\iff OF=DY \iff h_a=R\ .$

Remark. $h_a=R\iff bc=2R^2\iff \sin B\sin C=\frac 12$ . Prove easily that

$4R^2\left[\left(OX^2-AE^2\right)-\left(OY^2-AF^2\right)\right]=\left(h_a^2-R^2\right)\left(b^2-c^2\right)$ .


Proof 1 of PP1.2. Denote $A=2x$ , $B=2y$ , $C=2z$ $,\ \left(x+y+z=90^{\circ}\right)$ and $D'\in BC\cap w$ . Thus, $\left\|\begin{array}{c} \widehat{BB'Q}\equiv\widehat{CC'P}\\\ \widehat{BB'Q}\equiv\widehat{BD'P}\\\ \widehat{CC'P}\equiv\widehat{CD'Q}\end{array}\right\|$ $\implies$ $\left\|\begin{array}{c} \widehat{BB'Q}\equiv\widehat{CD'Q}\\\ \widehat{CC'P}\equiv\widehat{BD'P}\end{array}\right\|$

$\implies$ $\left\|\begin{array}{c} B'D'\parallel BI\\\ C'D'\parallel CI\end{array}\right\|$ and $BB'QD'$ and $CC'PD'$ are cyclically $\implies$ $\left\|\begin{array}{c} \widehat{D'BQ}\equiv\widehat{D'B'Q}\\\ \widehat{D'CP}\equiv\widehat{D'C'P}\end{array}\right\|$ . Prove easily that $m(\widehat{D'B'Q})=z$ and $m(\widehat{D'C'P})=y$ .

Therefore, $m(\widehat{D'BQ})=z$ and $m(\widehat{D'CP})=y$ , i.e. $CP\parallel IB\parallel B'D'$ and $BQ\parallel IC\parallel C'D'$ . In conclusion, the quadrilateral $IBMC$ is a parallelogram.

Proof 2 of PP1.2.

$ \blacktriangleright$ The point $ I$ is the orthocenter of the triangle $ I_aI_bI_c$ (evidently). Prove that the quadrilateral $ BPQC$ is cyclically.

Proof.
$ \blacktriangleright$ Denote $ T\in I_aM\cap BC$ . Show that $ \frac {TB}{TC} = \frac {p - c}{p - b}$ , i.e. $ \overline {I_aMT}\perp BC$ (the point $ T\equiv L$ and belongs to the circle $ w_a$ ).

Proof.
$ \blacktriangleright$ Apply lemma I to the triangle $ BCI_a$ , where $ I_aL\perp BC$ and for the point $ M\in I_aL$ the quadrilateral $ BPQC$

is cyclically. In conclusion, the point $ M$ is the orthocenter of $ \triangle BCI_a$ , i.e. $ CM\perp BI_a$ and $ BM\perp CI_a$ .

$ \blacktriangleright$ Apply lemma II to $ \triangle I_aI_bI_c$ with the orthocenter $ I$ , where $ M$ is the orthocenter of $ \triangle BCI_a$ .

In conclusion, $ ML$ is equally to the inradius of the orthic $ \triangle ABC$ w.r.t. the triangle $ I_aI_bI_c$ , i.e. $ ML = r$ .


Proof of PP2. Let $\begin{array}{c} R\in CP\cap AN\\\ S\in AP\cap CQ\end{array}$ . Since $BNRP$ is cyclically obtain $CN\cdot CB=CR\cdot CP\ (*)$ . Thus, $\boxed{AP\perp CQ}$ $\iff$ $Q$ is orthocenter

of $\triangle APC$ $\iff$ $AMRP$ is cyclically $\iff$ $CM\cdot CA=CR\cdot CP$ $\stackrel{(*)}{\iff}$ $CM\cdot CA=CN\cdot CB$ $\iff$ $ABNM$ is cyclically $\iff$ $\boxed{AB\perp AC}$ .


Proof of PP3. Denote the second intersection $S$ of $AD$ with $w$ and $G\in FD\cap CS$ . Since $HESC$ is a rhombus obtain

$DF=DG$ . From the "butterfly problem" appling to the concurrent chords $[AS]$ , $[BC]$ in $w$ obtain that $OD\perp FG$ , i.e. $DF\perp DO$ .


Proof of PP4. $\boxed{IN\perp MB}$ $\iff$ $IM^2-IB^2=NM^2-NB^2$ $\iff$ $IM^2-IA^2=NM^2-NB^2$ $\iff$ $MA^2=NM^2-NB^2$

$\iff$ $\boxed{MA^2+NB^2=MN^2}$ $\iff$ $MA^2+\left(IN^2-IB^2\right)=MN^2$ $\iff$ $MN^2-MA^2=IN^2-IA^2$ $\iff$ $\boxed{MI\perp AN}$ .
This post has been edited 84 times. Last edited by Virgil Nicula, Nov 23, 2015, 7:21 AM

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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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    by MrMustache, Nov 11, 2020, 4:52 PM

  • 372554 views!

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    by piphi, Jun 10, 2020, 11:44 PM

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    by CSPAL, May 27, 2020, 4:17 PM

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    by OlympusHero, May 14, 2020, 8:43 PM

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  • Joined: Jun 22, 2005
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  • Total entries: 456
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