# Is there a proof by “reverse construction”?

This is a question whether a proof I will describe is palatable.

I want to prove something, esp. a question on geometry, by assuming that which is in question to be in fact true, and then prove that it is absolutely probable to construct it geometrically that way.

For example, in this problem, it asks me to prove that $T,K$ and $M$ are collinear.

Let $C_1$ be a circle with centre $O$ and a chord $AB$,and consider a circle $C_2$ tangent internally to $C_1$ at $T$ and to $AB$ at $K$. Let $M$ denote the midpoint of the arc AB not containing $T$. Show that $T,K,M$ are collinear.

In the proof I submitted, I assume that $T,K,M$ are in fact collinear by drawing a line that passes through $T,K,M$ and showing that a possible geometric construction can be made, and that the circumstances and constraints of the problem are satisfied.

Suppose that we start by creating circle $C_1$ and it's chord $AB$. Now we find the midpoint of sector $ACB$ and label it $M$. Now we draw an arbitrary line that goes through $M$, $AB$ and circle $C_1$.

## The intersections will be $T,K$ and $M$ and thus by definition, they are and will be collinear.

Now we draw a line through $C_1$ and $T$. Then we draw a line perpendicular to line $C_1T$, and by definition, the perpendicular line will be tangent to $\bigcirc \ C_1$ at $T$. Since $\bigcirc C_2$ should be internally tangent to $\bigcirc C_1$ at $T$, then the center $C_2$ must necessarily lie on $C_1T$.

Now since $\bigcirc C_2$ must pass through, and in fact be tangent to, $T$ and $K$, then $TK$ will be a chord of $\bigcirc C_2$, and a perpendicular bisector of $TK$ at $C$ will pass through the center of $\bigcirc C_2$. Since we know the center of $\bigcirc C_2$ must be on line $C_1T$ then, the intersection of the perpendicular bisector of $TK$ and line $C_1T$ will be the center of $\bigcirc C_2$. This now begs the question as to why $\bigcirc C_2$ should be tangent to $AB$ at $K$. Notice that there a kite forms from the points $T,C_2,K,G$. Notice too that $C_2T \cong C_2K$, since they are the radii of $\bigcirc C_2$. Notice that $TC\cong KC$, because $C$ is the midpoint of $TK$. Because of these, $\angle C_2KG$ must be a reflection of $\angle C_2TG$. And through that, we prove that $K$ is indeed the point of tangency between $\bigcirc C_2$ and $AB$, and thus completing the proof that $T, K$ and $M$ are collinear.

Is this "proof by reverse construction" even allowed? Or am I just arbitrarily naming a generic process?

• of course, if you can draw a line thru $A$, $B$, $C$, they will be colinear by definition – Vasya Mar 27 '18 at 15:14
• Yes, so we have established that they are collinear. Then we will proceed with to "reverse construct" and show that the original construction is possible given that they are collinear. – John Glenn Mar 27 '18 at 15:17
• There is not a original construction that you need to prove is possible. You need to show that given the original construction, the points must be collinear. You have not done that. – Morgan Rodgers Mar 28 '18 at 23:32

## 2 Answers

If you have shown that $$A,B,C \text{ are colinear} \implies \text{the original contruction}$$

then, in general, you cannot conclude $$\text{the original contruction} \implies A,B,C \text{ are colinear}$$ as these are two different statements.

• Would it be the same if I arrive at the same exact construction? – John Glenn Mar 27 '18 at 15:45
• @JohnGlenn It's unclear what that would mean – Morgan Rodgers Mar 27 '18 at 15:54
• @MorganRodgers Maybe I meant: "the problem asks me to create this diagram, I follow the constraints and instructions, then I'm asked to prove, say, the points are collinear. But then for my proof, I begin with drawing a line and picking 3 points, so by definition they are in fact collinear. And from those 3 points, I trace back and create the diagram of the problem." Am I making sense? – John Glenn Mar 27 '18 at 16:02
• Sort of, but this answer exactly addresses that: No, you have not proven the statement. Proving that it is possible for the 3 points to be collinear is not the same as showing that the 3 points have to be collinear. If you are just trying to show that a specific diagram is possible, then you have constructed it and that's fine. That's different from proving a statement. – Morgan Rodgers Mar 27 '18 at 16:55
• @JohnGlenn it might help if you could post the exact question – jacer21 Mar 27 '18 at 17:22

I cannot entirely understand what you wrote, but I think that there's an idea there that might be illustrated by Conway's proof of Morley's Theorem, which says that if you take the two trisectors of each angle of a triangle, and intersect these in pairs (i.e., find the intersections of the trisectors 'adjacent' to each of the three edges), then the resulting intersections will form an equilateral triangle.

To quote the wikipedia article,

The [proof] starts with an equilateral triangle and shows that a triangle may be built around it which will be similar to any selected triangle.

In other words, the proof starts with the thing that is to be constructed, and then back-constructs all possible starting configurations.