SOLUTION: Given triangle ABC with D the midpoint of side AB and E on side BC, F on side AC and G is the intersection of side DE and side BF. BE:EC = 2:3 and DG:GE = 5:8. What is the ratio of

Question 1188287: Given triangle ABC with D the midpoint of side AB and E on side BC, F on side AC and G is the intersection of side DE and side BF. BE:EC = 2:3 and DG:GE = 5:8. What is the ratio of BG:GF?
Answer by yurtman(42) (Show Source): You can put this solution on YOUR website!
Certainly, let's find the ratio of BG:GF.
**1. Utilize Ceva's Theorem:**
* Ceva's Theorem states that for any triangle ABC, if lines AD, BE, and CF intersect at a single point (in this case, point G), then:
(AF/FC) * (BD/DA) * (CE/EB) = 1
**2. Apply the given ratios:**
* BD/DA = 1/1 (since D is the midpoint of AB)
* BE/EC = 2/3
**3. Calculate AF/FC:**
* Using Ceva's Theorem:
(AF/FC) * (1/1) * (3/2) = 1
AF/FC = 2/3
**4. Use Menelaus' Theorem:**
* Menelaus' Theorem states that for any transversal line (in this case, line DE) that intersects the sides of a triangle (triangle ABC) at points D, E, and F, then:
(AD/DB) * (BE/EC) * (CF/FA) = 1
**5. Apply the known ratios and solve for BG/GF:**
* (1/1) * (2/3) * (3/2) * (BG/GF) = 1
BG/GF = 1
**Therefore, the ratio of BG:GF is 1:1.**
Let me know if you have any other questions or problems to solve!

RELATED QUESTIONS
Triangle ABC is isosceles with AB = AC. Let D be the foot of the altitude from A on BC,... (answered by math_helper)

ABC is a triangle with base = BC . A line segment DE is drawn parallel to Side AC. D is... (answered by ikleyn)

In triangle ABC D and E are points on side AB and AC respectively DE||BC BD=5cm and... (answered by Edwin McCravy)

In a triangle ABC, D is a point on side AB and E is the point on AC. DE parallel to BC.... (answered by greenestamps)

A triangular lot ABC with BC=400m and B=50° is divided into two parts by the line... (answered by CPhill)

abc is a right triangle. is side ac=5 and side ab=10 what is the measure of side... (answered by Fombitz)

In triangle ABC, E is the midpoint of AC and D is the midpoint of CB. If DF is parallel... (answered by ikleyn)

In ∆ABC in the diagram below, D and E are points on side AB, and F and G are points on... (answered by lotusjayden,greenestamps)

in triangle abc, side ab = 17 inches and side bc = 25 inches. what could be the length of (answered by edjones)