The lengths of the sides of the triangle ABC are equal to 4, 5 and 6. Points A1, B1, C1

The lengths of the sides of the triangle ABC are equal to 4, 5 and 6. Points A1, B1, C1 are the bases of the heights of this triangle. Find the radius of the circle inscribed in the triangle A1, B1, C1.

To solve the problem, you need to know only two formulas:
r = ((p – a) * (p – b) * (p – c) / p) ^ 1/2;
But for this you need a perimeter, p = (a + b + c) / 2.
Let’s substitute:
p = (4 + 5 + 6) / 2;
p = 15/2;
p = 7.5;
Let’s substitute in the first formula:
r = ((7.5 – 4) * (7.5 – 5) * (7.5 – 6) / 7.5) ^ 1/2;
r = (3.5 * 2.5 * 1.5) / 7.5) ^ 1/2;
r = (131.25 / 7.5) ^ 1/2;
r = 17.5 ^ 1/2;
r = 31.02.
Answer: r = 31.02.