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The property of the median and height of an isosceles triangle

Lecture




Theorem

If the median in a triangle is also height, then such a triangle is isosceles.

The property of the median and height of an isosceles triangle

Evidence.

Let the triangle ABC CD be the median and height.
Then the triangles ACD and BCD are equal by the first sign of the equality of triangles, since CD is common for these triangles, AD = DB - because CD is the median and splits AB in half, ∠ CDA = CDB - because CD is height.
It follows that AC = CB and ABC is an isosceles triangle. The theorem is proved.

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Planometry

Terms: Planometry