Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, and C are 𝐚, 𝐛, and 𝐜 respectively, and the position vector of P is (𝐚 + 𝐛 + 𝐜)/4, then the position vector of the orthocentre of this triangle is:

O is orthocentre and G is centroid and C is circumcentre. G divides OC in 2:1 ratio. Position vector of centriod 𝐆 = (𝐚 + 𝐛 + 𝐜)/3 Position vector of circum center 𝐂 = (𝐚 + 𝐛 + 𝐜)/4 Apply Section Formula, 𝐆 = 2𝐂 + 𝐎/3 3𝐆 = 2𝐂 + 𝐎 𝐎 = 3𝐆 - 2𝐂 = (𝐚 + 𝐛 + 𝐜) - 2(𝐚 + 𝐛 + 𝐜)/4 = (𝐚 + 𝐛 + 𝐜) - (𝐚 + 𝐛 + 𝐜)/2 = (𝐚 + 𝐛 + 𝐜)/2