Find all right triangles with integer sides and inradius 6.
The formula for the inradius of a right triangle is 2r=a+b-c
12=a+b+sqrt(a^2+b^2)
b=(12a-72)/(a-12) (for a>12)
This is an integer for a={13,14,15,16,18,20} and repeating for (b,a,c)
Leading to the triples
(13,84,85)
(14,48,50)
(15,36,39)
(16,30,34)
(18,24,30)
(20,21,29)
(One for each factor of 12. Coincidence?)
A Desmos to see them all
https://www.desmos.com/calculator/z6ks0dge22
Edit: For the remark above. It's actually one for each pair of factors of 72= 2*6^2
https://www.sciencedirect.com/science/article/pii/S0022314X16301603?via%3Dihub
Edited on October 26, 2024, 5:37 pm
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Posted by Jer
on 2024-10-26 17:17:03 |
solution