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Strategy 10 of 12Auxiliary Construction
Create something new to solve the problem
GeometryExtra ElementsConnections
Add a new element to the problem — a line, point, function, or set — that wasn't in the original statement but helps connect the pieces and reach the solution.
When to use it
- →When the problem seems to be missing a connection
- →When you need an intermediate step
- →When an additional structure could bridge the elements
- →In geometric problems with missing connections
How to think (step by step)
- 1Identify the gap: What is missing to connect the elements?
- 2Think of the auxiliary: What extra element would help?
- 3Construct it
- 4Use it to solve the problem
- 5Verify the solution applies to the original problem
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
In triangle \(ABC\), let \(M\) be the midpoint of \(BC\). Prove the median length formula: \(AM^2 = \frac{2AB^2 + 2AC^2 - BC^2}{4}\).
📙Intermediate
Two circles intersect at \(A\) and \(B\). A line through \(A\) meets the circles at \(C\) and \(D\) (on the same line). Prove: \(\angle CBD\) is constant as the line through \(A\) varies.
📕Advanced
Let \(P\) be a point inside \(\triangle ABC\). Lines \(AP\), \(BP\), \(CP\) meet opposite sides at \(D\), \(E\), \(F\). Prove Ceva's Theorem: \(\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1\).