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Mathematics LibreTexts

13.2: Inradius of h-triangle

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Theorem 13.2.1

The inradius of any h-triangle is less than 12ln3.

Proof

Let I and r be the h-incenter and h-inradius of hXYZ.

Note that the h-angles XIY, YIZ and ZIX have the same sign. Without loss of generality, we can assume that all of them are positive and therefore

hXIY+hYIZ+hZIX=2π

截屏2021-02-24 下午3.30.51.png

We can assume that hXIY23π; if not relabel X, Y, and Z.

Since r is the h-distance from I to (XY)h, Proposition 13.1.1 implies that

r<12ln1+cosπ31cosπ3=12ln1+12112=12ln3.

Exercise 13.2.1

Let hABCD be a quadrangle in the h-plane such that the h-angles at A, B, and C are right and ABh=BCh. Find the optimal upper bound for ABh.

Hint

Note that the angle of prarllelism of B to (CD)h is bigger than π4, and it converge to π4 as CDh.

Applying Proposition 13.1.1, we get that

BCh<12ln1+12112=ln(1+2).

The right hand side is the limit of BCh if CDh. Therefore, ln(1+2) is the optimal upper bound.


This page titled 13.2: Inradius of h-triangle is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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