Final Project for Kanita DuCloux

Part A: I will 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. Then, explore (AF)(BD)EC) and (FB)(DC)(EA) for various triangles and various locations of P.

Observe the triangle below. (AF)(BD)EC) = (FB)(DC)(EA). Click on triangle to construct a triangle like the one below and drag a point to see that the two products are always equal.

Part B: Prove (AF)(BD)EC) = (FB)(DC)(EA)

To prove this, I need to draw some parallel lines to produce similar triangles and consider the ratio

and generalize to a point P outside the triangle.

Draw lines (BY and CY) parallel to AD. Now, triangles BPD and BXC are similar by AA AND

We also have triangles CPD similar to triangle CYB and thus

Taking the proportions above, we have

,

yields, (BD)(XC) = (BC)(PD) and (DC)(YB) = (PD)(BC). By substitution, we get

(BD)(XC) = (DC)(YB) or .

Next we have, triangle EXC similar to triangle EPA by AA. Going through a similar process, yields

We also have triangle FPA similar to triangle FYB by AA and

,,

Everything on the right-hand side of the equation cancels out and therefore equals 1. So,

Can this be generalized so that P is outside of the triangle? Observe the pictures below. There are a lot of lines in the pictures but look at the distances. It appears that the same is true if the point P is outside.

Then, click outside to try it yourself.

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

Observe the following picture. Then, click on area and drag a vertex to see what happens.

When is the ratio equal to 4? Look and see. Point P is the centroid.
The text passage describes a mathematical exploration and proof related to a geometric configuration involving a triangle and a point P, both inside and outside the triangle. It includes a specific mathematical question about the ratio of the areas of two triangles and when this ratio equals 4.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Final Project for Kanita DuCloux

Part A: I will 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. Then, explore (AF)(BD)EC) and (FB)(DC)(EA) for various triangles and various locations of P.

Observe the triangle below. (AF)(BD)EC) = (FB)(DC)(EA). Click on triangle to construct a triangle like the one below and drag a point to see that the two products are always equal.

Part B: Prove (AF)(BD)EC) = (FB)(DC)(EA)

To prove this, I need to draw some parallel lines to produce similar triangles and consider the ratio

and generalize to a point P outside the triangle.

Draw lines (BY and CY) parallel to AD. Now, triangles BPD and BXC are similar by AA AND

We also have triangles CPD similar to triangle CYB and thus

Taking the proportions above, we have

,

yields, (BD)(XC) = (BC)(PD) and (DC)(YB) = (PD)(BC). By substitution, we get

(BD)(XC) = (DC)(YB) or .

Next we have, triangle EXC similar to triangle EPA by AA. Going through a similar process, yields

We also have triangle FPA similar to triangle FYB by AA and

,,

Everything on the right-hand side of the equation cancels out and therefore equals 1. So,

Can this be generalized so that P is outside of the triangle? Observe the pictures below. There are a lot of lines in the pictures but look at the distances. It appears that the same is true if the point P is outside.

Then, click outside to try it yourself.

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

Observe the following picture. Then, click on area and drag a vertex to see what happens.

When is the ratio equal to 4? Look and see. Point P is the centroid.
user
The text passage describes a mathematical exploration and proof related to a geometric configuration involving a triangle and a point P, both inside and outside the triangle. It includes a specific mathematical question about the ratio of the areas of two triangles and when this ratio equals 4.

TYPE: SINGLE_QUESTION_WITH_ANSWER