In triangle ABC with right angle C, the median CM is 10 roots of 3. The inscribed circle in triangle ACM

In triangle ABC with right angle C, the median CM is 10 roots of 3. The inscribed circle in triangle ACM touches the hypotenuse AB at point P. Find leg BC if AP / PB = 1/3

It is necessary to remember:
median drawn from the right angle of the wound to half of the hypotenuse;
the segments of the tangents drawn from a point outside the circle to the points of tangency with the circle are equal;
Pythagorean theorem – the sum of the squares of the legs is equal to the square of the hypotenuse.
Let’s get started:
AB = 2CM = 20√3;
CM = AM – ∆AMС – isosceles and the radius of the inscribed circle lies on the median MK (KС = KA);
РA = 1/4 AB = 1/4 * 20√3 = 5√3;
РA = AK = 5√3;
AC = 10√3;
BC = √ (AB ^ 2 – AC ^ 2) = √ ((20√3) ^ 2 – (10√3) ^ 2) = √ (1200 – 300) = √900 = 30.