Construct a triangle where , angle , the distance from down to the line segment is and is closer to than .
Draw an auxiliary figure:
You start by constructing the line . Then you construct perpendicular lines from both and and set the height to cm.
You must now find a way to construct , and it’s here you use the circle of Apollonius. The only thing you know is that is supposed to be on the dotted line that is cm from , but you don’t know where.
On the other hand, you do know from the circle of Apollonius that if you make a circle that goes through and , the center of the circle forms the central angle to any inscribed angle that spans . Therefore you can use that to find the angles and . You can use Formula ():
Thus, you construct ° angles at and so that they turn inwards. The intersection point of these angle rays is the center of the circle of Apollonius.
Since and are both °, the angle .
This fits with being the central angle of . Thus, you can draw the circle with center in and radius . You find where the circle crosses the dotted line. This happens at two points, but the task tells you to choose the one closest to . Thus, you can draw the lines between , and so that the triangle is complete.
Isn’t it beautiful?