6. Stages of Solving Construction Problems

 Problem 16.  Construct a triangle, given c, h_c , m_c.

Analysis

Let the triangle ABC be the one we are looking for. AB = c, therefore the construction of the segment АB is done.

CH = h_c , therefore the point C lies on a line parallel to AB and at a distance h_c  from it.

CM = m_{c ,} therefore the point C lies on a circle with center in the midpoint of AB and radius m_c.

Then C is the intersection point of the line and circle you constructed.

Construction

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Proof

 AB = c

C lies on the line g (or f) , therefore the altitude through the vertex C has length h_c .

C lies on the circle k, therefore the median through the vertex C has length m_c .

The triangle ABC is the one we are looking for.

Exploration

The points A and B are defined unequivocally.

Point С is the intersection point of lines and a circle.

If h_c < m_c, then the circle intersects each of these two lines in 2 points. Then the problem has 4 solutions.

If h_c = m_c, then the circle touches each line. Then the problem has 2 solutions.

If h_c >m_c, then the circle does not intersect the lines and the problem has no solution.

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