Find a right triangle and an isosceles triangle that have same perimeter and area.

All sides are integers.

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- April 18th 2012, 06:51 AMWilmerright=isosceles
Find a right triangle and an isosceles triangle that have same perimeter and area.

All sides are integers. - April 18th 2012, 08:03 AMbiffboyRe: right=isosceles
Any right angled isosceles triangle!

- April 18th 2012, 08:35 AMPlatoRe: right=isosceles
- April 18th 2012, 08:39 AMbiffboyRe: right=isosceles
Sorry, I'll have start reading the whole question!

- November 26th 2012, 08:04 PMPaddyMacRe: right=isosceles
- November 26th 2012, 08:11 PMPaddyMacRe: right=isosceles
- November 26th 2012, 08:13 PMPaddyMacRe: right=isosceles
Oops, I need to read to whole question. The perimeters are different.

- November 26th 2012, 09:42 PMMacstersUndeadRe: right=isosceles
only a start to a possible solution...

__Spoiler__: - November 27th 2012, 12:20 AMWilmerRe: right=isosceles
Hint: the isosceles triangle is made up of 2 pythagorean congruent right triangles "stuck together"

along one of the legs; so you need to find 2 right triangles: one has area = twice the other's area.

If isosceles triangle made up of 2 right triangles ABC (BC=a, AC=b, AB=c), sharing leg AC,

and other right triangle is DEF (EF=d, DF=e, DE=f), then:

2a + 2c = d + e + f (perimeter)

ab = de / 2 (area) - November 27th 2012, 01:00 AMMarkFLRe: right=isosceles
You may generate as many such right triangles as you like by taking two successive members of the Pell sequence as "seeds" for Euclid's method for generating Pythagorean triples.

Here are the first few examples (after the two already given):

(119,120,169), (696,697,985), (4059,4060,5741), (23660,23661,33461). - November 27th 2012, 03:48 AMWilmerRe: right=isosceles
Or looping c from 5 increment by 2: a = [SQRT(2c^2 - 1) - 1] / 2

Next 4:

137903, 137904, 195025

803760, 804761, 1136689

4684659, 4684660, 6625109

27304196, 27304197, 38613965