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Problem 275

# $$\mathrm{D}$$ and $$\mathrm{E}$$ are respective points of side $$\underline{\mathrm{AB}}$$ and $$\underline{\mathrm{BC}}$$ of $\triangle \mathrm{ABC}$$, so that$$\mathrm{AD} / \mathrm{DB}=2 / 3$$and$$\mathrm{BE} / \mathrm{EC}=1 / 4 .$$If$$\underline{\mathrm{AE}}$$and$$\underline{D C}$ meet at $$\mathrm{P}$$, find $$\mathrm{PC} / \mathrm{DP}$$.

Expert verified
The ratio PC/DP is 1:1, meaning the segments PC and DP have equal lengths.
See the step by step solution

## Step 1: Understand the problem and label the triangle

Draw triangle ABC with points D on AB and E on BC. Mark point P as the intersection of lines AE and DC. We are given the following ratios: AD/DB = 2/3 and BE/EC = 1/4. Our goal is to find the ratio PC/DP.

## Step 2: Identify similar triangles

To solve this problem, we will need to identify similar triangles within the figure. Notice that triangles ADP and EDP are similar because they share angle D and angle ADE is congruent to angle EDE. Additionally, triangles BEP and ECP are similar because they share angle E and angle DEC is congruent to angle ECB.

## Step 3: Determine ratios of corresponding sides for similar triangles

As triangles ADP and EDP are similar, we can write the following side ratios: $\frac{AP}{EP} = \frac{AD}{ED}$ And for triangles BEP and ECP, $\frac{BP}{EP} = \frac{BE}{EC}$

## Step 4: Use given ratios to express ratios in terms of AD and BE

We are given that AD/DB = 2/3 and BE/EC = 1/4. Therefore, $ED = AD + DB = AD + \frac{3}{2}AD = \frac{5}{2}AD$ And, $EC = BE + CD = BE + DB + DB = BE + 2DB = BE + (3DB - BE) = 3DB = 3(\frac{3}{2}AD) = \frac{9}{2}AD$ Substitute the values of ED and EC in the ratios found in step 3: $\frac{AP}{EP} = \frac{AD}{\frac{5}{2}AD} = \frac{2}{5}$ $\frac{BP}{EP} = \frac{BE}{\frac{9}{2}AD} = \frac{2}{9}$

## Step 5: Apply Ceva's theorem

Ceva's theorem states that for point P inside triangle ABC, lines AP, BP, and CP are concurrent (intersect at a single point) if and only if the ratio product: $\frac{AP}{PB} \cdot \frac{BP}{PC} \cdot \frac{CP}{PA} = 1$ Using the values found in Step 4, we can determine the ratio PC/DP: $\frac{AP}{EP} \cdot \frac{DP}{PC} \cdot \frac{CP}{AP} = 1$ $\frac{2}{5} \cdot \frac{DP}{PC} \cdot \frac{CP}{AP} = 1$

## Step 6: Solve for PC/DP ratio

To find the value of PC/DP, we need to manipulate the equation from Step 5: $\frac{DP}{PC} = \frac{5}{2} \cdot \frac{AP}{CP}$ Use the equation found in step 4: $\frac{DP}{PC} = \frac{5}{2} \cdot \frac{2}{5}$ $\frac{DP}{PC} = 1$ Therefore, the ratio PC/DP is 1:1, meaning the segments PC and DP have equal lengths.

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