Some Lemmas on Cyclic Quadrilaterals

by AlastorMoody, Nov 23, 2018, 7:33 PM

We will first check out some basic properties of a cyclic quadrilateral,

If points $A,B,C,D$ lie on circle $\omega $, then $ABCD$ is a cyclic quadrilateral
For proving a quadrilateral cyclic, proving any of the following will prove the desired quadrilateral cyclic,
Criteria 1#
Criteria 2#


Lemma 1:
If side $BA \cap CD= P$, then, $AD$ is $ /|$ to $BC$ in $\Delta PBC$, or $\Delta PAD \sim \Delta PCB$

proof

Lemma 2:
Power of Point Theorem: If side $BA \cap CD= P$, then, $PA \cdot PB =PD \cdot PC$

proof
Some Advantages:

$\text{Power of point theorem helps formation of similar triangles with SAS similarity,}$ $\text{ and when There are two or more powers of a single circle,}$ $\text{ it often makes multiple pairs of similar triangles which }$ $\text{sometimes have something similar between them }$ $\text{ and that helps as an aid for angle chasing as well as length chasing,}$ $\text{ Don't miss those similar triangles !!}$

Lemma 3:
Power of point (in disguise):If $PT$ is tangent to $(ABCD)$ at $T$, then, $PT^2=PA \cdot PB=PD \cdot PC$

proof

Lemma 4:
$ABCD$ is such a cyclic quadrilateral, that, $AC$ is a diameter, and if $K$ is foot of the altitude from $A$ to $BD$,then, $\angle DAC=\angle KAB$

proof(1)
proof(2)

Lemma 5:
If a cyclic quadrilateral is a trapezium, it has to be an isosceles trapezium

proof

Lemma 6:
If $H$ is the orthocenter of $\Delta ABC$, then $H$ is the incenter of the orthic triangle of $\Delta ABC$

proof

Lemma 7:
If $H$ is the orthocenter of $\Delta ABC$, then reflection of $H$ over any side of $\Delta ABC$ lies on circumcircle of $\Delta ABC$

proof

Lemma 8:
If $H_M$ be the reflection of orthocenter $H$ of $\Delta ABC$ over mid-point $M$ of side $BC$, then, $H_M$ lies on $(ABC)$ and $AH_M$ is the diameter

proof

Lemma 9:
Let $\Delta ABC$ be inscribed in a circle with center $O$ and Let line $\mathcal{L}$ be tangent at $C$, and let two points on either side of $C$ be $X$ and $Y$ that lie on line $\mathcal{L}$, such, that $X$ is closer to $A$ than $B$ and $Y$ is closer to $B$ than $A$, Then,
$\angle B=\angle ACX$ and $\angle A=\angle BAY$

proof

Lemma 10:
Let $\Delta ABC$ be inscribed in circle $\omega $, Let angle bisector $AD \cap \omega =X$, such $D \in BC$, Prove,
$\text{(i) }$ $BX =CX$
$\text{(ii) }$ If $\mathcal{L}$ is the perpendicular bisector of $BC$, then $X$ lies on $\mathcal{L}$

proof

Lemma 11:
Let $\triangle ABP$ with $M$ on the midpoint of $PB$ and $MB$ is tangent to $(ACB)$,such that $C \in AM$, then $MP$ is tangent to $(ACP)$.

proof



Problems:

1# JBMO ShortList 2015 G1 Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A,B,D,E$ belong to the same circle.

2# (own :D) The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$, Prove, $PC=2MC$

3# JBMO ShortList 2015 G2 The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.

4# Indian RMO 1992 P4 Let $ABCD$ be a quadrilateral inscribed in circle $\omega $, such that $AC \perp BD $ and $AC \cap BD=E$, If $R$ is radius of $\omega $, then prove,
$$AE^2+BE^2+CE^2+DE^2=4R^2$$
5# USAMO 2003 P4 Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.

6# ISL 2010 G1 Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$

7# Evan Chen's EGMO Let $ABCD$ be a cyclic quadrilateral, Let $I_1$ be incenter of $\Delta ABC$ and Let $I_2$ be the incenter of $\Delta BDC$, prove, that $I_1I_2CB$ is also cyclic quadrilateral

8# USAJMO 2011 P5 Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that
(i) lines $PB$ and $PD$ are tangent to $\omega$,
(ii) $P, A, C$ are collinear, and
(iii) $DE \parallel AC$.
Prove that $BE$ bisects $AC$.

9# Source: Unknown A circle $(I)$ touches the sides $AB,BC,CD,DA$ of a quadrilateral $ABCD$ at $E,F,G,H$ respectively. $EG$ intersects $FH$ at $P$. The line passing through $P$ and perpendicular to $IP$ intersects $AD,BC$ at $K,L$ respectively. Prove that if $EG \perp FH$ then $HK=FL$.

10# Indian RMO 1999 P3 Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.

11# Indian RMO 2000 P5 The internal bisector of angle $A$ in a triangle $ABC$ with $AC > AB$ meets the circumcircle $\Gamma$ of the triangle in $D$. Join$D$ to the center $O$ of the circle $\Gamma$ and suppose that $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.

12# Source:Unknown Let $ABCD$ is $||^{gm}$, such $\angle BAC \leq 90 $. Let $P$ is point on $BD$ such that $\angle PCB = \angle ACD $. The circumcircle of $\Delta ABD$ cut $AC$ at $E$, show that, $\angle AED= \angle PEB$ .

13# JBMO Shortlist 2011 G2 Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

14# IMO 1990 P1/ ISL 1990 P11 Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If
\[ \frac {AM}{AB} = t, \]find $\frac {EG}{EF}$ in terms of $ t$.

15# Canada MO 1999 P3 Let $P$ be a point inside the circle $\omega$. Consider the set of chords of $\omega$ that contains $P$. Prove that their midpoints all lie on a circle.

16# IMO 2006 P1 Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\]Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

17# USAMO 1999 P6 Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$. The inscribed circle $\omega$ of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\angle DAC$ such that $EF \perp CD$. Let the circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$. Prove that the triangle $AFG$ is isosceles.

I guess :D, these problems are enough for warm-up/practice, others can be added in the comments section!
This post has been edited 45 times. Last edited by AlastorMoody, Jan 15, 2019, 5:45 PM

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Don't forget IMO 2006/1 :o

by Kagebaka, Dec 23, 2018, 8:25 PM

I'll talk about all possible non-sense :D

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  • what a goat, u used to be friends with my brother :)

    by bookstuffthanks, Jul 31, 2024, 12:05 PM

  • hello fellow moody!!

    by crazyeyemoody907, Oct 31, 2023, 1:55 AM

  • @below I wish I started earlier / didn't have to do JEE and leave oly way before I could study conics and projective stuff which I really wanted to study :( . Huh, life really sucks when u are forced due to peer pressure to read sh_t u dont want to read

    by kamatadu, Jan 3, 2023, 1:25 PM

  • Lots of good stuffs here.

    by amar_04, Dec 30, 2022, 2:31 PM

  • But even if he went to jee he could continue with this.

    Doing JEE(and completely leaving oly) seems like a insult to the oly math he knows

    by HoRI_DA_GRe8, Feb 11, 2022, 2:11 PM

  • Ohhh did he go for JEE? Good for him, bad for us :sadge:. Hmmm so that is the reason why he is inactive
    Btw @below finally everyone falls to the monopoly of JEE :) Coz IIT's are the best in India.

    by BVKRB-, Feb 1, 2022, 12:57 PM

  • Kukuku first shout of 2022,why did this guy left this and went for trashy JEE

    by Commander_Anta78, Jan 27, 2022, 3:42 PM

  • When are you going to br alive again ,we miss you

    by HoRI_DA_GRe8, Aug 11, 2021, 5:10 PM

  • kukuku first shout o 2021

    by leafwhisker, Mar 6, 2021, 5:10 AM

  • wow I completely forgot this blog

    by Math-wiz, Dec 25, 2020, 6:49 PM

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    by DuoDuoling0, Dec 22, 2020, 10:54 PM

  • This site would work faster if not all diagrams were displayed on the initial page. Anyway I like your problem selection taste.

    by WolfusA, Sep 24, 2020, 7:58 PM

  • nice blog :)

    by Orestis_Lignos, Sep 15, 2020, 9:09 AM

  • Hello everyone, nice blog :)

    by Functional_equation, Sep 12, 2020, 6:22 PM

  • pro blogo

    by Aritra12, Sep 8, 2020, 11:17 AM

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