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Show that a triangle is right angled

Posted: Sat Apr 20, 2019 3:00 pm UTC
by kaloyan_kolev
In △ABC (not isosceles) CH, CL and CM are respectively height, bisector and median. Show that ∠ACB=90 degrees if and only if ∠HCL=∠MCL.

I think that I have to show that △MCB (or △ACM) is isosceles, but I can't figure it out. I will be very grateful if you help me.

Re: Show that a triangle is right angled

Posted: Sun Apr 21, 2019 12:18 pm UTC
by Qaanol
Height of what, bisector of what, and median of what?

Try drawing a picture.

Is this homework?

Re: Show that a triangle is right angled

Posted: Sun Apr 21, 2019 4:16 pm UTC
by ucim
Assuming H, L, and M are on segment AB (which is the only way to make sense of the question), ask yourself the following questions:

1: Sum of ∠A, ∠B, and ∠C is.... what?
2: What does that imply if (as desired) ∠C = 90 degrees?
3: Considering all the angles at C (except ∠ACB)... what is their sum?
4: Consider the consequences of the congruent angles of the two triangles you suspect must be isosceles.

That should get you going.

Jose