152. Concurrence, cyclically, metrics, symmedian a.s.o.

by Virgil Nicula, Oct 11, 2010, 12:15 PM

Quote:
Proposed problem. Let $D$ be an inner point w.r.t. the triangle $ABC$ for which $\left\|\ \begin{array}{c} \widehat {DAB}\equiv\widehat {DBC}\\\\ \widehat {DAC}\equiv\widehat {DCB}\end{array}\right\|$ . Prove that :

1. The line $AD$ pass through the midpoint $M$ of the side $[BC]$ .

2. If $E$ is the symetrical point of $A$ w.r.t. the point $M$ , then the quadrilateral $BDCE$ is cyclically.

3. If denote the circumcircles $w_1=C(O_1,r_1)$ , $w_2=C(O_2,r_2)$ of the triangles $ADB$ and $ADC$ respectively, then exists the relation $\frac {r_1}{r_2}=\left(\frac cb\right)^2$ .

4. If the lines $EB$ , $EC$ meet again the circles $w_1$ , $w_2$ in the points $U$ , $V$ respectively, then $A\in UV$ .

5. The ray $[EA$ is the $E$-symmedian in the triangle $UEV$ .

6. .............

7. .............
This post has been edited 4 times. Last edited by Virgil Nicula, Nov 22, 2015, 9:14 PM

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