⚖️ Mass Points Templates

Mass point geometry provides an elegant alternative to coordinate methods for ratio problems. Master these templates for AMC 12 success.

🎯 Recognition Cues

Key Triggers

  • Ratio problems - Look for problems asking for ratios of lengths
  • Cevians - Look for lines from vertices to opposite sides
  • Concurrency - Look for lines meeting at a point
  • Weighted points - Look for problems involving weighted averages

Common Setups

  • Triangles with cevians
  • Ratio problems
  • Concurrency problems
  • Weighted point problems

🧩 Solution Template

Step 1: Assign Masses

  • Assign masses to vertices proportional to opposite side lengths
  • Use the principle of moments
  • Ensure masses are consistent

Step 2: Find Center of Mass

  • Calculate the center of mass
  • Use the formula: $G = \frac{m_1P_1 + m_2P_2 + m_3P_3}{m_1 + m_2 + m_3}$

Step 3: Apply Mass Point Properties

  • Use the fact that center of mass is at intersection of cevians
  • Apply mass ratios to find unknown lengths
  • Use weighted averages

Step 4: Verify

  • Check that masses are consistent
  • Ensure ratios make geometric sense
  • Verify the final answer

🔍 Worked Example

Problem: In triangle $ABC$, $D$ is on $BC$ such that $BD:DC = 3:4$. If $AB = 6$ and $AC = 8$, find the ratio $AD:DE$ where $E$ is the intersection of $AD$ and the line through $B$ parallel to $AC$.

Solution: Step 1: Assign masses

  • Mass at $B$: 4 (opposite to $AC$)
  • Mass at $C$: 3 (opposite to $AB$)
  • Mass at $A$: 7 (sum of masses at $B$ and $C$)

Step 2: Find center of mass

  • Center of mass is at $D$ (since $BD:DC = 3:4$)
  • Mass at $D$: 7 (sum of masses at $B$ and $C$)

Step 3: Apply mass point properties

  • Since $E$ is on line through $B$ parallel to $AC$, we can use similar triangles
  • $\triangle BDE \sim \triangle CDA$ by AA
  • Ratio of similarity: $\frac{BD}{DC} = \frac{3}{4}$
  • Therefore, $AD:DE = 4:3$

Step 4: Verify

  • Check that masses are consistent ✓
  • Ratio makes geometric sense ✓
  • Answer is correct ✓

Answer: $AD:DE = 4:3$

⚠️ Common Pitfalls

Pitfall: Wrong mass assignment

  • Fix: Masses are proportional to lengths of opposite sides

Pitfall: Forgetting about center of mass

  • Fix: Center of mass is at intersection of cevians

Pitfall: Wrong mass ratios

  • Fix: Use mass ratios consistently

Pitfall: Forgetting about similar triangles

  • Fix: Look for similar triangles in mass point problems

🔗 Related Patterns

💡 Quick Reference

Mass Point Rules

  • Mass assignment: Proportional to opposite side lengths
  • Center of mass: At intersection of cevians
  • Ratio finding: Use mass ratios

Common Applications

  • Cevians: Finding ratios of cevian segments
  • Concurrency: Proving concurrency of cevians
  • Ratios: Finding ratios of lengths

Solution Strategy

  • Identify: Look for ratio problems with cevians
  • Assign: Assign masses to vertices
  • Apply: Use mass point properties
  • Calculate: Solve for unknown ratios

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