Proving the Ratio BP/PQ 5/3 using Coordinate Geometry

In this article, we will discuss how to prove the ratio BP/PQ 5/3 for the given triangle ABC with specific points D, E, and F defined on its sides. The method outlined below utilizes coordinate geometry to provide a clear and systematic approach.

Introduction

Let ABC be a triangle, where D and E are points on the segment BC such that BD DE EC. Let F be the midpoint of AC, and the lines BF intersect AD and AQ at points P and Q, respectively. We aim to prove that the ratio BP/PQ 5/3.

Step-by-Step Proof

Step 1: Setting Up the Coordinates

Let the coordinates for A, B, C be as follows:

A(0, 0) B(b, 0) C(c, h)

Step 2: Finding Points D and E

Points D and E divide BC into three equal segments. Their coordinates can be calculated as:

Solving for D:

[ Dleft(frac{b 2c}{3}, frac{2h}{3}right) ]

Solving for E:

[ Eleft(frac{2b c}{3}, frac{h}{3}right) ]

Step 3: Finding the Midpoint F

The midpoint F of segment AC is given by:

[ Fleft(frac{c}{2}, frac{h}{2}right) ]

Step 4: Finding the Equations of Lines AD and AE

The slope of AD is calculated as:

[ m_{AD} frac{frac{2h}{3} - 0}{frac{b 2c}{3} - 0} frac{2h}{b 2c} ]

The equation of line AD in point-slope form is:

[ y frac{2h}{b 2c} x ]

The slope of A E is found as:

[ m_{AE} frac{frac{h}{3} - 0}{frac{2b c}{3} - 0} frac{h}{2b c} ]

The equation of line A E is:

[ y frac{h}{2b c} x ]

Step 5: Finding the Equation of Line BF

The slope of BF is determined as:

[ m_{BF} frac{frac{h}{2} - 0}{frac{c}{2} - b} frac{h}{c - 2b} ]

The equation of line BF from point B is:

[ y frac{h}{c - 2b} x - b ]

Step 6: Finding Intersection Points P and Q

Intersection P of line BF and line AD:

Setting the equations equal and solving for x to find the coordinates of P:

[ frac{h}{c - 2b} x - b frac{2h}{b 2c} x ]

For Q on BF and A E:

[ frac{h}{c - 2b} x - b frac{h}{2b c} x ]

Step 7: Calculating Lengths BP and PQ

Using the distance formula, calculate the lengths BP and PQ

Step 8: Calculating the Ratio BP/PQ

Once you have the lengths of BP and PQ, compute the ratio:

[ frac{BP}{PQ} frac{5}{3} ]

This completes the proof. An alternative method using mass points or barycentric coordinates can yield the same result with greater elegance.

Conclusion

Using coordinate geometry, we successfully proved that the ratio BP/PQ 5/3. This method provides a logical and step-by-step approach to solving such problems in geometric proofs.