In triangle ABC, the segment joining the orthocentre and the circumcentre is parallel to side BC.
Statement 1: tan A, tan B and tan C are in arithmetic progression.
Statement 2: If tan A, tan B and tan C are in arithmetic progression, then tan A × tan C = 3.
Choose the correct option:
(a)

Question

In triangle ABC, the segment joining the orthocentre and the circumcentre is parallel to side BC.
Statement 1: tan A, tan B and tan C are in arithmetic progression.
Statement 2: If tan A, tan B and tan C are in arithmetic progression, then tan A × tan C = 3.
Choose the correct option:
(a) Statement 1 is true, Statement 2 is true; Statement 2 correctly explains Statement 1
(b) Statement 1 is true, Statement 2 is true; Statement 2 does not correctly explain Statement 1
(c) Statement 1 is false, Statement 2 is true
(d) Statement 1 is true, Statement 2 is false

Accepted Answer

c. OH parallel to BC requires equal heights above BC: R cosA = 2R cosB cosC, which simplifies to tanB tanC = 3 (NOT that tanA,tanB,tanC are in AP), so Statement 1 is false. Statement 2 is true: if tanA,tanB,tanC are in AP then tanA+tanB+tanC = 3tanB, and since this sum equals tanA tanB tanC, we get tanA tanC = 3. Hence Statement 1 false, Statement 2 true = option (c).

Solution

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