Line Triangle
Why does the incenter and the circumcenter lie on the same line in an isosceles triangle?
My question exactly...
I don't understand your question. The incenter and circumcenter are two points and any two points can be joined by a line, so yes they do lie on the same line. Maybe your question is why do they both lie on the median to the third side (not one of the congruent sides)?
If that is indeed your question, then consider the symmetry of an isosceles triangle. Specifically the median drawn to the third side is also an angle bisector and an altitude. The incircle must lie on this angle bisector (since it lies at the intersection of all 3 angle bisectors) and the circumcenter must lie on this line as well since it is also an altitude and the circumcenter lies at the intersection of all 3 altitudes. Thus the incenter and circumcenter of an isosceles triangle both lie on the median to the third side.
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