Equilateral Triangle with Orthocenter at Origin

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x + 2√2y = 4.

If the coordinates of the vertex A are (α, β), then the greatest integer less than or equal to |α + √2β| is ________.

Correct Answer: 4

Explanation

In an equilateral triangle, the orthocenter, centroid and circumcenter coincide. Since the orthocenter is at the origin, the centroid is also at the origin.

Distance of origin from the line x + 2√2y − 4 = 0 is:

= |−4| / √(1 + 8) = 4 / 3

In an equilateral triangle, the distance of a vertex from the opposite side is three times the distance of the centroid from that side.

So, distance of vertex A from side BC = 3 × (4/3) = 4

Using distance formula for point (α, β) from the line:

|α + 2√2β − 4| / 3 = 4

⇒ |α + 2√2β − 4| = 12

This gives possible values of |α + √2β|, whose greatest integer value is 4.

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