The Plemelj Construction of a Triangle: 9

This Demonstration constructs a triangle given the length of the base , the length of the altitude from to and the difference of the angles at and . This construction unifies two constructions mentioned in The Plemelj Construction of a Triangle: 1.
Step 1: Draw a line segment of length and a perpendicular line segment of length with midpoint .
Step 2: Draw a circle with center such that subtends an angle from points on above the chord . The angle equals .
Step 3: Find a point on at distance from and a point on at distance from .
Step 4: Draw the isosceles trapezoid .
Step 5: The point is the intersection of the straight line through parallel to and the perpendicular bisector of and .
Step 6: The triangle meets the stated conditions.
The three triangles , and are congruent. In the isosceles triangle , , so .
On the other hand, .
So and .
The pink, blue and green arcs have measures , and .


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For the history of this problem and references, see The Plemelj Construction of a Triangle: 1.
This construction is similar to that in The Plemelj Construction of a Triangle: 7.
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