Some basic properties of incircles-excircles for beginners

by AlastorMoody, Nov 21, 2018, 6:32 PM

Lemma 1:
If the incircle of $\Delta ABC$ touches $BC, CA, AB$ at $A_1, B_1, C_1$, then, $A_1C_1 \perp BI$, $A_1B_1 \perp CI$, $B_1C_1 \perp AI$

proof(1)
proof(2)

Lemma 2:
If the incircle of $\Delta ABC$ touches $BC, CA, AB$ at $A_1, B_1, C_1$, then, $AI$ bisects $B_1C_1$

proof
Note

Lemma 3:
If $C_1B_1$ is extended to point $N$, such that, angle bisector $BI$ meets $C_1B_1$ at $N$, then, $\angle BNC= 90^{\circ}$

proof
Note

Lemma 4:
If $\Delta ABC$ is a triangle with incenter $I$ and let $D, E, F$ be the tangency points of the incircle with $BC, CA, AB$ respectively, then,the lines $ID$ and $EF$ intersect on the $A$-median of triangle $\Delta ABC$.

proof

Lemma 5:
If $AE=AF$, such, $E,F \in AB,AC$ respectively, and Let $AD$ be the angle bisector, then, $AD$ bisects $EF$

proof

Lemma 6 (An aid to angle chasing):
If $I$ is the incenter of $\Delta ABC$, then, $\angle BIC=90^{\circ}+\frac{1}{2}\angle A$

proof

Lemma 7 (Incenter-Excenter Lemma)
Let $I$ be the incenter of $\Delta ABC$ and Let $AI$ intersect $(ABC)$ at $D$, and Let $AD \cap (BIC)=X$ and Let $I_A$ be the excenter produced opposite to vertex $A$, then:
$\text{(i) } BD=CD=ID$
$\text{(ii) } BX$ and $ AX $ bisect exterior angle of $ \angle B \& \angle C$ respecitvely
$\text{(iii) } A-I-I_A $

proof

Lemma 8
Let $\Delta DEF$ be the contact triangle of $\Delta ABC$, and let $DI \cap EF =X$, then $AX$ bisects $BC$

proof

Lemma 9
Let $I$ and $I_A$ be the incenter and excenter of $\Delta ABC$, $\Delta DEF$ be the contact triangle and $DI$ intersects $(I)$ at $E \in (I)$, $AE \cap BC =X$, Let $H_M$ be the mid-point of $AH_A$ and $BM=CM$, then:
$\text{(i) }$ $AE||IM$
$\text{(ii) }$ $H_M - I -X$
$\text{(iii) }$ $H_M - D -I_A$

proof

Lemma 10
In $\Delta ABC$, $D$ is point of contact of incircle on $BC$ and $F$ is the midpoint of $BC$. If $I$ is the incentre, prove that, $FI$ extended bisects $AD$.


proof


Problems:

1# Spanish Mathematical Olympiad 1991 The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$

2# Indian RMO 2018 P6 Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that,

$\text{(i)}$ $B,C,Y,X$ are concyclic.
$\text{(ii)}$ $I$ is also the incentre of triangle $DYX$.

3# Indonesia Round 2 2012 The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

4# FBH- Regional Olympiad 2012 Let $S$ be an incenter of triangle $ABC$ and let incircle touch sides $AC$ and $AB$ in points $P$ and $Q$, respectively. Lines $BS$ and $CS$ intersect line $PQ$ in points $M$ and $N$, respectively. Prove that points $M$, $N$, $B$ and $C$ are concyclic

5# JBMO ShortList 2007 G3 Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at $N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such that $\frac{DP}{DN}=\frac{BD}{CD}$ . Show that $DM \perp PN$ .

6# JBMO 2016 P1 A trapezoid $ABCD$ ($AB || CD$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

7# CGMO 2012 P5 The in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$.

8# 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 $BD$.

9# Evan Chen's EGMO The incircle of $ABC$ is tangent to $\overline{BC}, \overline{CA}, \overline{AB}$ at $D, E, F$, respectively. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AC}$, respectively. Ray $BI$ meets line $EF$ at $K$. Show that,
$\text{(i) }\overline{BK} \perp \overline{CK}$.
$\text{(ii) } K$ lies on line $MN$.
$\text{(iii) } BM= MK$

10# 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.

11# JBMO SL 2014 G1 Let ${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.

12# Indian RMO 2010 P5 Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.

13# Indian RMO 2011 P5 Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$

I guess :D, these problems are enough for warm-up/practice, others can be added in the comments section!
This post has been edited 48 times. Last edited by AlastorMoody, Dec 25, 2019, 6:46 AM

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There is a typo in lemma 9
$DI$ meets $(I)$ at $E$

Thanks! Corrected!
This post has been edited 1 time. Last edited by AlastorMoody, Dec 25, 2019, 6:46 AM

by SHREYAS333, Dec 24, 2019, 11:03 AM

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

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    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

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