Final Assignment

By Nikki Masson

Part I: Consider any triangle ABC. Select a point P inside the triangle and draw lines AP, BP, and CP extended to their intersections with the opposite sides in points D, E, and F respectively.

Now move the point P around and explore the lengths of AF, BD, CE, EA, FB, and DC.

A. Calculate:

B. Move the point P and calculate the same relationship:

C. Move the point P one more time and again calculate the relationship:

Conjecture: When P is inside the triangle the relationship (AF*BD*CE)/(EA*FB*DC)=1.

Part II: Now we will prove the conjecture we just made.

This conjecture we just made is called Ceva's Thoerem and it states that:

If the points F, D and E are on the sides AB, BC and AC of a triangle then the lines AD, BE and CF are concurrent if and only if the product of the ratios

Proof:

1.) Extend the lines BE and CF beyond the triangle and draw a line through A and parallel to BC. Mark the points where the the extended lines cross the parallel line.

2.) There are several pairs of similiar triangles which give us the following ratios:

a.) EBC and EAY are similiar triangles

So we get the ratio: (EC/EA)=(BC/YA)

b. FBC and FAX are similiar triangles

So we get the ratio: (FA/FB)=(AX/BC)

c. XAP and CDP are similiar triangles

So we get the ratio: (XA/CD)=(PA/PD)

d. BDP and YAP are similiar triangles

So we get the ratio: (BD/YA)=(PD/PA)

If we mulitply the ratios together, we get:

(BD/YA)*(XA/CD)*(EC/EA)*(FA/FB)=(PD/PA)*(PA/PD)*(BC/YA)*(AX/BC)

Then when we simplify we get:

(BD/CD)*(XA/GA)*(EC/EA)*(FA/FB)=(AX/GA)

Next mulitple both sides by (GA/AX)

Then our final formula is:

(BD/CD)*(EC/EA)*(FA/FB)=1!

We have proved our conjecture.

What would happen if P is outside the triangle? You can explore by using the GSP Sketch.

Part III: Show that when P is inside the triangle ABC, the ratio of the areas of the triangle ABC and triangle DEF is always greater than or equal to 4. When is it equal to 4?

What if we move point P?

Let's move point P one more time?

Now let's explore why our conjecture that the relationship between these two triangles is greater than or equal to 4.

The medial triangle DFE is 1/4 of the area of the triangle ABC, so we have a ratio of 4 to 1.

When does the ratio equal 4 exactly? Use the link and explore the relationship when P is on the centriod.

GSP Sketch