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The Plemelj Construction of a Triangle: 9

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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.

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Construction

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.

Verification

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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Contributed by: Izidor Hafner, Nada Razpet and Marko Razpet (September 2017)
Open content licensed under CC BY-NC-SA

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Details

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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