home cabri The incircle
unexpected properties
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To prove
The triangle ABC is right-angled so that AB²+BC²=AC². DE, DF and DG are all radii of the inscribed circle. We will prove that when AB, BC and AC are all integers, then the radius is an integer also.
The proof
1. DE = BE = BG(because angle DBE is 45 degrees)
2. EC = BC - DE
3. FC = BC - DE
4. AF = AC - (BC-DE)
5. AG = AC - (BC-DE)
6. AG = AB - DE
7. AC - (BC - DE) = AB - DE
8. AC - BC = AB - 2DE
9. DE = (AB+BC-AC)/2

This means that DE is a linear combination of AB, BC and AC. If they are all integers then (AB+BC-AC) is an integer too. Thus DE will be an integer as long as (AB+BC-AC) is even.
If AB and BC are both even, or AB and BC are both odd, then because AB²+BC²=AC², clearly AC² (and AC) must be even and therefore the above result follows.
If one of AB or BC is even and the other is odd then AB²+BC² must be odd and therefore AC² (and AC) must be odd. (AB+BC-AC) will then be even.

So, DE is always an integer. Some examples:
AB BC AC radius of incircle
3 4 5 1
6 8 10 2
5 12 13 2