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Problem 67 from Kiselev's Geometry,

Prove that in an isosceles triangle, two medians are congruent, two angle bisectors are congruent, two altitudes are congruent.

Here is the content page in the book. The problem belongs to Chapter 1.5.

The theorems covered in chapter 1.5 are,

(i) In an isosceles triangle, the bisector of the angle at the vertex is at the same time the median and the altitude.

(ii) In an isosceles triangle, the angles at the base are congruent.

(iii) In an isosceles triangle, the bisector of the angle at the vertex is an axis of symmetry of the triangle.

I have understood the proof for these theorems and I have tried to use these theorems and others mentioned in previous chapters to prove the theorem in problem 67, but I can't prove it.

Please help. Also, please avoid using theorems that are covered in the later part of the book. :)

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    $\begingroup$ Are you talking about angle bisectors ?? $\endgroup$ – Anik Bhowmick Sep 20 '18 at 9:05
  • $\begingroup$ Have you tried similarity ?? $\endgroup$ – Anik Bhowmick Sep 20 '18 at 9:11
  • $\begingroup$ Okay, I have done it with the help of similarity on triangles. $\endgroup$ – Anik Bhowmick Sep 20 '18 at 9:21
  • $\begingroup$ In each case, you need to prove triangles congruent, then use the fact that corresponding parts of congruent triangles are congruent to draw the desired conclusion. $\endgroup$ – N. F. Taussig Sep 20 '18 at 10:03
  • $\begingroup$ The congruency tests are what I had in mind. I am not familiar with the book, so I do not know what results you are allowed to use. $\endgroup$ – N. F. Taussig Sep 20 '18 at 12:12
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First thing to do is make a picture.

enter image description here

We need to prove each of these separately.

In all three cases, though the proof will be based on the idea that $\triangle APB \cong \triangle AQC$

We can prove one by SAS

One by ASA

While we cannot generally prove congruence by AAS, we can when they are right triangles.

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  • $\begingroup$ I have managed to prove that line segments $AP$ and $BQ$ are congruent. But it was first assumed that the three sides of the triangle are congruent. I cannot have this assumption to prove the theorem for triangles that have exactly two congruent sides. $\endgroup$ – yh05 Sep 21 '18 at 2:12
  • $\begingroup$ @yh05 is that better. $\endgroup$ – Doug M Sep 21 '18 at 2:27
  • $\begingroup$ Seems like we have to use the idea of congruent triangles. It's a little weird to me because the author presented the chapter on congruence triangles after this set of problems. =/ $\endgroup$ – yh05 Sep 21 '18 at 2:31
  • $\begingroup$ You could make an argument based on the axis of symmetry. But I think that is higher level thinking than the congruent triangle arguments. $\endgroup$ – Doug M Sep 21 '18 at 3:10

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