Circles Within Triangles

[Graeme]

Here's a funny thing I only just discovered today, don't laugh, because you already knew, but, since I was 12 I've known about Pythagoras' theorem, the law of a right angled triangle and the 3, 4, 5 triangle.

But today I found out that if you put a circle inside a 3, 4, 5 triangle with a diameter such that it touches all three sides of the triangle, then the circle's area is Pi!

Awesome!

Graeme

[HansV]

A right-angled triangle whose sides are whole numbers is called a Pythagorean triangle, and the length of its sides is called a Pythagorean triple.
3-4-5 is the smallest Pythagorean triple, others are 5-12-13 and 6-8-10 (the latter is simply double 3-4-5).
Fun fact: the radius of the inscribed circle of a Pythagorean triangle (the circle touching each of the sides) is always a whole number.

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[BobH]

Those are things I never knew and I made A's in geometry.

If the smallest Pythagorean triangle will contain a circle of area = pi, what would be the area of the largest Pythagorean triangle that is twice the size of the smallest? Is it 4 times larger?

Interesting that the radii change as they do with the increased size of the triangles sides.

[Graeme]

Fun fact: the radius of the inscribed circle of a Pythagorean triangle (the circle touching each of the sides) is always a whole number.

Another mathematical wow on the same day!

Thanks

Graeme

[HansV]

If the smallest Pythagorean triangle will contain a circle of area = pi, what would be the area of the largest Pythagorean triangle that is twice the size of the smallest? Is it 4 times larger?

Yes - the area increases by the square of the length of the sides.