Find the distance from point K to side BC

Liudmyla Hetmanenko
The circle touches AB and AC the lateral sides of the isosceles triangle ABC at the vertices B and C.
On the arc of this circle, which lies inside this triangle, there is a point K so that the distances from it to the sides AB and AC are equal to 24 cm and 6 cm appropriately.
Find the distance from point K to side BC.

Construction of a figure with the conditions imposed in the statement:

Let's take a point K in the plane. Let two points E1 and E2 be such that KE1=6 and KE2=24.
The perpendicular (r) through E1 to KE1 and the perpendicular (s) through E2 to KE2 intersect at A.
We construct the two circles (KBC) and (KB'C'), solution of the Apollonian Problem (PRR), which pass through K and are tangent to (r) and a (s).
Let F and F' be the orthogonal projections of K onto BC and B'C', respectively. We have KF=KF' and this distance, 12, does not depend on the location location of points E1 and E2.

Angel Montesdeoca. May, 2024