abbracciopj

2022-06-14

Finding the slopes of internal or external angle bisectors, when given the slopes of the sides of a triangle, it's a simple question, the one you can easily find in any textbook of analytic geometry.

But the reverse problem : finding the slopes of the sides of a triangle, when only given the slopes of internal angle bisectors. Is it possible? Is there a straightforward formula for that?

candelo6a

Expert

Yes. The slopes of the internal bisectors intersect at the incentre I. You will thus know the angles AIB, BIC and CIA.

180 = AIB + IAB + IBA

180 = ACB + 2*IAB + 2*IBA

ACB = 2*AIB - 180

From this you can figure out the side slopes.

Yahir Crane

Expert

First step: Knowing the slopes of the bisectors you calculate tan(A/2), tan(B/2), tan(C/2).
Second step: you calculate the slopes of straight lines from the vertices making these angles to the bisectors.
Case closed.

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