Triangle Constructions
Drew Cronic

Constructing triangles given certian pieces of information about them can be an interesting activity. I enjoy it because it's a lot like putting a puzzle together. Sometimes these constructions can be very difficult and even frustrating, but to me, that just makes it even more of a challenge. When you finally solve the construction, you feel a real sense of accomplishment.
Let's start out with some relatively simple constructions.
1. We'll construct a triangle given two of its sides, say side a and side b, and an angle opposite one of the sides, say angle F.

First copy one of the sides, say side a, and copy angle F onto side a.

Now construct a circle with radius equal to side b and center on the left endpoint of side a.

Now construct a segment from the center of the circle to the intersection between the circle and the ray adjacent to side a.

Now we have our triangle.

To see GSP sketch click HERE

2. Now let's try to construct a triangle given two sides, say side a and side b, and the altitude to the third side, say altitude h(c).

First let's construct a line with a perpendicular segment of length h(c).

Now we can construct two circles with center at the top of h(c), one with radius of length side a, the other with radius of length side b.

Construct side a and side b from the center to the intersection points of the circles and the original line.

We now have the desired triangle.

To see GSP sketch click HERE

3.) Here's a more challenging one. Construct a triangle given one side, say side a, the median to side a, say m(a), and another median, say m(b).

First construct a segment congruent to side a and its midpoint.

In geometry there is a theorem that states that a triangle's medians all intersect at a point that divides each median into two segments with a ratio of 2:1 with the longer segment intersecting the triangle at a vertex. In other words, the medians intersect each other at their trisection points. So let's trisect m(a), and m(b).

Now we can construct two circles, one with center at the midpoint of side a and radius 1/3 of m(a). The other with center on one of side a's endpoints and radius 2/3 of m(b).

Construct the two medians through the intersection piont of the two circles.

We can now construct segments from the end of m(a) to the endpoints of side a.

Now we have our triangle.

To see GSP sketch click HERE

4.) Construct a triangle given its three medians, say m(a), m(b), and m(c).

First, let's construct a triangle with three sides equal to these three medians.

Now let's trisect the red segment, which is actually m(a), and then construct lines through that point parallel to the green and blue segments.

We can use the same theorem that we used on our last construction that says all the medians of a triangle intersect at a point that divides the medians into two segments with a ratio of 2:1. Let's trisect m(b) and m(c).

Now construct two circles, one with ridius 2/3 of m(b) and one with radius 2/3 m(c), with centers at m(a)'s trisection point.

Now find the intersection points between the line and the circle of its color farthest from the right endpoint of m(a), because they will be vertices.

These two new points are the final two verteces of our triangle. Now we can construct the sides to our triangle.

So we now have the desired triangle with the given medians.

To see GSP sketch click HERE

5.) Construct a triangle given two of its sides, say side a and side b, and the median to the third side, say m(c).

First construct side a and its midpoint.

Now construct a circle with radius of length m(c) and center at the left endpoint of side a. Construct another circle with radius 1/2 of side b and center at the midpoint of side a. These two circles will intersect at the midpoint of the unknown side.

Now we can construct m(c) from the left endpoint of side a to the intersection of the two circles.

Construct a ray from the right endpoint of side a through the endpoint of m(c), and construct a circle with radius of length side b and center at the left endpoint of side a.

Construct a segment from the left endpoint of side a to the intersection between the ray and the blue circle.

Now we have the desired triangle.

To see GSP sketch click HERE

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