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In this investigation, we will prove that this construction is actually a parabola by using properties of congruent triangles.

Parabola Construction

Steps:

(1) Construct a line (the directrix) and a point not on the line (the focus). In order to investigate the properties of the parabola, let's also construct a point on the line and label it "Drag Me".
(2) Construct the perpendicular bisector of the segment from the directrix to the focus. (Note: This line will be important later!)
(3) Construct the perpendicular that intersects the directrix at the Drag Me point. Label the intersection of this line and the line constructed in (2) as point P.
(4) Construct the locus of points (???) as the Drag Me point moves along the directrix. You will create a parabolic shape.
Is this REALLY a parabola?? Or is it an imposter?? Lets prove that it is a parabola by using properties of congruent triangles. Note: We will use letters to label our constructions at this point.
As stated in step (2), Line PM is the perpendicular bisector of segment XF. Therefore, segment XM is congruent to segment MF, and angle PMX is congruent to angle PMF.
If we construct the segment PF, we now have two triangles. Both triangles share the same side (segment PM). The Side-Angle-Side Congruence Axiom states that if two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent. Segments PX and PF are congruent as a result of this, therefore proving that this construction is in fact a parabola.

We have just proven that we have constructed a parabola by using properties of congruent triangles.

Now let's explore more!!

Remember in Step (2) when it was mentioned that the perpendicular bisector we constructed would be important later?? Well here's why:

That perpendicular bisector is tangent at point P on the parabola. Click HERE to explore on GSP what happens to that tangent line if we move the Drag me point (also known as point X) along the directrix.

Suggestions: Click on Animate Point at different times so that you can observe what is going on in the construction. What is changing? What is constant? Is it what you expected?

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