construct an inscribed circle in a circular sector.Explain each step
I hope this doesn't come too late...
I assume that you have a circular sector.
1. Construct the angle bisector passing through S. The intersection of the angle bisector and the arc of the sector is Z. Z must be a tangent point of the circle in question.
2. Choose an arbitrary point $\displaystyle M_a$ on the angle bisector which is the midpoint of the arbitrary circle $\displaystyle c_a$ with the radius $\displaystyle r = |\overline{M_a Z}|$ (in blue)
3. Draw a line through $\displaystyle M_a$ perpendicular on one leg of the sector. This line intersects the circle $\displaystyle c_a$ in $\displaystyle T_a$
4. Draw a line $\displaystyle ZT_a$ which intersect the leg of the sector in T. This is a tangentpoint of the circle c which you are looking for.
5. A parallel to $\displaystyle M_aT_a$ through T intersects the angle bisector in M which is the midpoint of the circle c. (in red)
Actually the construction uses similar triangles (indicated by different patterns in red) and the point Z as the fixpoint of a central dilation.
posted by bjhopper
reply by earboth
thanks much for your clever method. Iwas stuck but before your reply had worked it out differently.I drew in the chord then its perpendicular bisector.I extended the sector legs outside the arc and drew a parallel to the chordmeetind the arc at the intersection of arc and perpendicular. the new larger isosceles trangle has two outside equal angles. The bisector of one of them meets the original perpehdicular bisector at the center of the inscribed circle
bj