If the lengths of the two right sides of a right triangle are 3 and 4 respectively, what is the radius of its inscribed circle

If the lengths of the two right sides of a right triangle are 3 and 4 respectively, what is the radius of its inscribed circle

The radius of the inscribed circle is equal to (3 + 4-5) / 2 = 1

If the ratio of the three sides of a triangle is 3:4:5, what is the radius ratio of the inscribed circle to the circumscribed circle

Let three sides be 3k, 4K and 5K
It is easy to prove that a triangle is a right triangle
Inscribed circle radius = 1 / 2 (3K + 4k-5k) = k
Circumcircle radius = 1 / 2 * 5K = 2.5k
So the radius ratio of inscribed circle to circumscribed circle is 1:2.5 = 2:5

The mathematical problem "what is the ratio of the radius of the inscribed circle to the radius of the circumscribed circle of a triangle whose three sides are 3, 4 and 5 respectively"

Let AB = 5, BC = 4, AC = 3, the center of circumscribed circle be the middle point of hypotenuse AB (the center line on hypotenuse is equal to half of hypotenuse AB) and the radius be half of hypotenuse AB 5 / 2. Let the center of inscribed circle be o and the radius be r. triangle ABC can be divided into three parts: AOB, BOC, AOC and r

It is proved that the inner and outer centers of equilateral triangle coincide, and the radius of circumscribed circle is twice that of inscribed circle
How to prove it?
emergency

Let the equilateral triangle ABC cross point a as ad perpendicular to BC, vertical point d cross point B as be perpendicular to AC, vertical point ead intersects with be, connects CF at point F, and extends CF intersection AB to G