Kiselev's geometry Problem 67: In an isosceles triangle, two medians/bisectors/altitudes are congruent

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.

• Are you talking about angle bisectors ?? Sep 20, 2018 at 9:05
• Have you tried similarity ?? Sep 20, 2018 at 9:11
• Okay, I have done it with the help of similarity on triangles. Sep 20, 2018 at 9:21
• 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. Sep 20, 2018 at 10:03
• 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. Sep 20, 2018 at 12:12

First thing to do is make a picture.

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.

• 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.
– yh05
Sep 21, 2018 at 2:12
• @yh05 is that better. Sep 21, 2018 at 2:27
• 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. =/
– yh05
Sep 21, 2018 at 2:31
• You could make an argument based on the axis of symmetry. But I think that is higher level thinking than the congruent triangle arguments. Sep 21, 2018 at 3:10

This problem comes before the theorems on triangle congruence are introduced. An earlier commenter says, "you could make an argument based on the axis of symmetry", and in fact this is exactly what Kiselev intends: half the chapter is devoted to proving various properties of isosceles triangles, and the other half is devoted to explaining the concept of 'axial symmetry' and showing that (in Kiselev's words):

in an isosceles triangle, the bisector of the angle at the vertex is an axis of symmetry of the triangle.

Following this, Kiselev remarks:

Two symmetric figures can be superimposed by rotating one of them in space about the axis of symmetry until the rotated figure falls into the original plane again.

Remembering that congruence is defined in terms of superposition, it is easy to understand the proofs Kiselev was trying to lead his reader towards: By rotating an isosceles triangle around its axis of symmetry, we carry one median onto another (because the sides are congruent), one angle bisector onto another (because the base angles are congruent), and one altitude onto another (because the perpendicular drawn from a point to a line is unique (§24)).