Nice Geometry Problem

The incenter of $\triangle ABC$ touches its sides at $D,E,F$ respectively, and $BE, CF$ intersect the incircle second time at $M, N$ respectively. Extend $MD$ to $P$ so that $DP=2MD$ . Prove that $DE$ is tangent to the circle $\odot (DPN)$ .

#Geometry

Easy Math Editor

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Comments

More incircle properties!

Property1: $\frac {DN}{EN}=\frac {DF}{EF}$ .

Proof: Since $CD,CE$ are tangent to the incircle, $\triangle CDN\sim \triangle CFD\implies \frac {DN^2}{DF^2}=\frac {CN}{CF}$ . Symmetrically we have $\frac {EN^2}{EF^2}=\frac {CN}{CF}$ , thus $\frac {DN}{DF}=\frac {EN}{EF}$ and switching $DF,EN$ gives us our desired equality.

Property2: Let $L$ denote the midpoint of $EM$ , then $\triangle DML\sim \triangle DEF$

Proof: Note that $\angle ILB=\angle IFB=\angle IDB=90^{\circ}$ , therefore $D,I,L,F,B$ are concyclic where $I$ is the incenter of $ABC$ . Hence $\angle DLM=\angle DFB=\angle DEF$ . Since $\angle DML=\angle DFE$ by cyclic, we have our desired similarity by AA

Correlary: $\frac {DN}{EN}=\frac {DF}{EF}=\frac {DM}{ML}=\frac {2DM}{EM}$

Main proof: It suffices to show $\angle DPN=\angle NDE=\angle NME$ , since $\angle PDN=\angle NEM$ by cyclic, we will prove that $\triangle NDP\sim \triangle NEM$ . This is equivalent to $\frac {DN}{EN}=\frac {DP}{EM}=\frac {2DM}{EM}$ by SAS similarity, which follows from the correlary above.

Note: A cyclic quadrilateral that satisfies property 1 is called harmonic quadrilateral, and it is intimately related to concepts such as harmonic division and poles and polar.

Xuming Liang - 5 years, 6 months ago

BTW did u get this from AoPS? I just went on and saw the exact same problem

Xuming Liang - 5 years, 6 months ago

Yes I did :)

Alan Yan - 5 years, 6 months ago

Btw do you have a document that goes over some incircle properties? I am not that familiar with them and they will be very useful.

Alan Yan - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$