Euclidia isosceles triangle on tangent points 3.10

Question

I've been struggling to come up with a solution for the Euclidia problem 3.10, creating an isosceles triangle from two given points on a circle. I can do it, but not in the # of steps given in the hints, for both the L and E solutions. Web search only find solutions up to 3.8. Has anyone here figured these out? Maybe provide another hint? Thanks

Answer 1

For L: Draw the line that pases for the bottom point and the centre of the circunference, then draw the mediatrix between the two given points. Draw the perpendicular to the bottom point(tangent to the circunference) and where this line cross with the mediatrix, draw the line that passes from that point and from the other given point, last step is intuitive.

For E: Lets call the centre of the circunference C, the top given point A and the bottom given point B.

1-Draw the circunfence with centre on B and radius BA.It intersects the first circunference on points A and C.

2-Circunference with centre on C and radius CA.It intersects the last circunference on A and D.

3- Draw the line CD

4-Draw the circunference with centre on A and radius AB. It intersects the original circunference on two points, lets the one on the right be E.

5-Draw the circunference with centre on B and radius AE. It intersects the one created on step 4 on points E and F.

6-Draw the segment BF. It crosses the line CD on the point G.

7-Circunference with centre on B and radius BG. Intersets with BF on G and H.

8- Draw the segment HA.