# How to draw a triangle

Kapitoly: Triangle, The height of a triangle, The weight of a triangle, Circles in a triangle, Right triangle, How to draw a triangle, Area of the triangle, The Pythagorean Theorem

A triangle is a polygon that has exactly three sides. We denote these sides by lowercase letters, classically a, b, c. A triangle also has three vertices, which we denote by capital letters, classically A, B, C. Side a then corresponds to the line segment BC, side b to the line segment AC, and side c to the line segment AB. Side a is always opposite the vertex A, likewise for the other sides.

## We know all three sides

If we know the lengths of all three sides, drawing a triangle is quite easy. Pick one of the sides, say AB, and draw it as a normal line segment. Then we need to find out where the third point is. This is easily done by taking a compass, poking it into the point B and drawing a circle with a side length BC, say five centimetres. This tells us wherever the point C might be, so that it is just five centimetres away from the point B. Then we do the same with the side AC and the point A. Now we have two circles, and their intersection gives us a point that is a specified number of centimetres away from the vertex A and also from the point B.

## We know the two sides and the height

Each side of a triangle has a height, which is defined as the perpendicular from the opposite point to the specified side. If the perpendicular does not directly intersect the side, we must stretch the side to where it intersects the perpendicular. Thus, the basic property of height is that it is perpendicular to the side to which it belongs. This is often used in calculus problems because we can use the Pythagorean theorem. The height per side C is usually written using a lower case v and then in the subscript the label of the side to which the height belongs. Thus, in our case v_{c}. The heel of the perpendicular (the intersection of the side with the perpendicular) is denoted similarly, except that instead of a lowercase v, we write an uppercase P: P_{c}.

We have drawn the height from the vertex C. The other point is on the opposite side of AB (or side c), and the line CP_{c} is perpendicular to the side c. The point P_{c} is called the base of the height; the side c is called the base. We usually name the heel after the letter P with a subscript where the vertex from which the height leads is. In this case, it is the vertex C.

Heights are pretty easy to draw; you take a ruler and draw a perpendicular line from the side c so that this perpendicular line just intersects the point C. That's it.

We can draw a height from each vertex of the triangle. All the altitudes then intersect at a point called the orthocenter. The orthocenter may or may not lie inside the triangle. In the case of an acute triangle, the orthocentre lies inside the triangle:

## We know the two sides and the center of gravity

In addition to height, every triangle has three lines of gravity. The center of gravity is the line that connects the vertex to the center of the opposite side. All three lines of gravity intersect at a single point, which is called the center of gravity. This centre of gravity is then always divided by the lines of gravity in the ratio 2:1 - the part of the line of gravity in front of the centre of gravity is either twice as small or larger than the other part behind the centre of gravity. The center of gravity is named with the lower case letter t and with the subscript of the state to which it belongs. The centre of gravity is then usually named with one capital letter T.

If you know the two sides and the center of gravity to either of those two sides, you can draw a triangle very easily. You start by drawing the side AB again, and then find where the point C lies by making two circles, the first centered at the point B (or A, if you know the length AC) and the second centered at the midpoint of the side AB. The point where the circles intersect is the point C. I don't think the animation is needed anymore, the procedure is the same as in the first animation, except that instead of drawing the triangle ABC, you first draw the triangle S_{c}BC, from which you find the vertex C.

## We know two sides and a different center of gravity

First, look at the sketch to see what it might look like:

We know the items in red and we have to draw the whole triangle according to them. The procedure is as follows -- we complete the triangle on the quadrilateral and then use the quadrilateral to draw the rest of the triangle. This is clearly explained in the following figure (recall that the length of BD is twice the length of t_{b}).

This quadrilateral is easy to draw. First we draw the triangle ABD. We know the length of the side AB, the length of the side BD is 2t_{b}, and the length of AD is the same as the length of BC. We can find the point C either by finding the point T_{b}, which is half of the line BD, and then just draw the half line AT_{b}, and the length of AC is the length of 2AT_{b}. The other option is to draw a line parallel to the side AB and the point C will lie on this line and will be |AB| away from the point D. The following video shows this much more clearly: