210. Tangential circle of the triangle ABC.

by Virgil Nicula, Jan 18, 2011, 7:01 PM

Proposed problem. Let $ ABC$ be a triangle which is touched by its incircle in $ D\in (BC)$ , $ E\in (CA)$ , $ F\in (AB)$ .

Denote $ K\in BE\cap DF$ , $ L\in CF\cap DE$ and the midpoints $ M$ , $ N$ of $ [KE]$ , $ [LF]$ respectively. Prove that $ \widehat {EFM}\equiv\widehat {FEN}$ .

Proof 1 (metric). $ \frac {FD}{FE} = \frac {\sin\widehat {FED}}{\sin\widehat {FDE}}=\frac {\sin\left(90^{\circ}-\frac B2\right)}{\sin\left(90^{\circ}-\frac A2\right)}=$ $\frac {\cos\frac B2}{\cos\frac A2}=\sqrt {\frac {b(p- b)}{a(p- a)}}$ $ \implies$ $ \boxed {\ \frac {FD}{FE} = \sqrt {\frac {b(p - b)}{a(p - a)}}\ }$ . Denote $ S = [ABC]$ .

$ \frac {KF}{KD}\stackrel{1^*}{\ = \ } \frac {EA}{EC}\cdot\frac {BF}{BD}\cdot\frac {BC}{BA} = \frac {p - a}{p - c}\cdot\frac ac$ $ \implies$ $ \frac {KF}{a(p - a)} = \frac {KD}{c(p - c)} = \frac {FD}{a(p - a) + c(p - c)}$ $ \implies$ $ \boxed {\ KF = \frac {a(p - a)}{a(p - a) + c(p - c)}\cdot FD\ }$ .

Denote $ x = m(\widehat {EFM})$. Observe that $ m(\widehat {KFM}) = 90^{\circ} - \frac C2 - x$ . Therefore, $ MK = ME$ $ \stackrel{2^*}{\implies}$ $ KF\cdot\sin\widehat {KFM} = FE\cdot\sin\widehat {EFM}$ $ \implies$

$ KF\cdot\cos\left(\frac C2 + x\right) =$ $ FE\cdot\sin x$ $ \implies$ $ \frac {\sin x}{\cos\left(\frac C2 + x\right)} =$ $ \frac {KF}{FE} =$ $ \frac {a(p - a)}{a(p - a) + c(p - c)}\cdot\frac {FD}{FE} =$ $ \frac {a(p - a)}{a(p - a) + c(p - c)}\cdot\sqrt {\frac {b(p - b)}{a(p - a)}} =$

$ \frac {\sqrt {ab(p - a)(p - b)}}{a(p - a) + c(p - c)}$ $ \implies$ $ \frac {\tan x}{\cos\frac C2 - \sin\frac C2\cdot\tan x} =$ $ \frac {\sqrt {ab(p - a)(p - b)}}{a(p - a) + c(p - c)}$ $ \implies$ $ \tan x = \frac {S}{a(p - a) + c(p - c) + (p - a)(p - b)}$ $ \implies$

$ \tan x = \frac {4S}{4a(b + c) - 3a^2 - (b - c)^2}$ , which is a symmetrical form w.r.t. the pair $ (b,c)\ \implies\ \boxed {\ \tan x = \tan y = \frac {4S}{4a(b + c) - 3a^2 - (b - c)^2}\ }$ .

Lemma 1*.

Lemma 2*.

Proof 2 (synthetic).
This post has been edited 4 times. Last edited by Virgil Nicula, Nov 22, 2015, 4:14 PM

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