Point Solution: If point P is the circumcenter, then it
is equidistant to all of the vertices of triangle ABC, so AP=BP=CP. Therefore,
all of the interior triangles are isosceles, so their base angles are
congruent. So if the measure of angle ABP is 40 degrees, then so is angle BAP.
Also, since the measure of angle CBP
is 30 degrees, then so is angle PCB. Since all of the angles of a triangle add up
to 180 degrees, then angle APB
measures 100 degrees and angle BPC
measures 120 degrees. Since there are
360 degrees in a circle, the measure of angle APC is 140 degrees. Angles PAC and PCA have to be congruent and add up to 180 degrees along with angle
APC, so the measure of angle PAC is 20 degrees. |