Thread: URGENT Geometry - triangle (angle bisector, median, altitude)

1. URGENT Geometry - triangle (angle bisector, median, altitude)

Prove that a triangle in which the median and the altitude is symmetric with respect to the angle bisector from the same vertex must have a right angle in this vertex. (Please take a look at the image attached.)

I can prove that if the triangle is right-angled, then the symmetry holds but I can't prove that if it isn't right-angled then they aren't symmetric.

2. Reply

AD = DB = DC = radius
DB = DC gives us CDB = 180 - 2 * BCD
AD = DB gives us ADC = 180 - 2 * ACD
CDB + ADC = 180
thus
90 = ACD + BCD

3. Yeah, I know that but what if $\overline{DC}\ne r$. Sorry for the misleading figure - $C$ is not on the circle (if it were then the triangle would be right-angled - Thales' theorem).

4. Reply

There is always a circle passing by three points. If it is not this one it's gonna be the other one?

5. Yes but $\overline{AD}\ne \overline{BD}$. I mean the circumcenter won't be $D$, the midpoint of $\overline{AB}$.

6. Reply

Yeah, if I get something I tell you.

7. Thank you very much.