In this post we continue the series on triangles. As promised in the previous post, we will derive formulas related to the various centers of a triangle. If you wish to refresh your mind about the centers of the triangle, please refer to Nomenclature for Properties of Triangles. Now, each center is a point located relative to the triangle. This means that each center has distances from each of the sides and vertices of the triangle. In some cases, there is a line from the vertex to the opposite side that passes through a center and this line must have a length. In what follows, we will derive these formulas.
The Circumcenter – Distance from Vertices (Circumradius)
With reference to the figure above, O is the circumcenter of ΔABC nd D, E, F are the feet of the perpendiculars from O onto BC, CA, AB respectively.
Hence by definition of the circumcircle, AO = BO = CO = R.
The Circumcenter – Distance from Sides
With reference to the same figure above ∠BOC = 2∠CAB = 2A
In ΔBOD and ΔCOD, OB = OC, OD = OD, ∠ODB = ∠ODC = 90°
Hence by RHS criterion ΔBOD ≡ ΔCOD we get ∠BOD = ∠COD = A
Hence OD = OB cos ∠BOD
Similarly,
The Centroid – Length of Medians
With reference to the figure above, G is the centroid of ΔABC.
D, E, F are the midpoints of sides BC, CA, AB respectively.
Hence AD, BE, CF are the medians of ΔABC.
Let AD = x
The Centroid – Distance from Vertices
We know that the centroid divides a median in the ratio 2:1, the greater portion being the distance between the vertex and the centroid. Hence, we get
The Incenter – Distance from Sides (Inradius)
With reference to the figure alongside, I is the incenter of ΔABC.
X, Y, Z are the feet of the perpendiculars from I onto the sides BC, CA, AB respectively.
Now by definition of the inradius IX = IY = IZ = r.
Now, using Δ to symbolize the area of a triangle, we have ΔABC = ΔIBC + ΔICA + ΔIAB
Also,
The Incenter – Distance from Vertices
With reference to the same figure above
Similarly,
The Orthocenter – Distance from Vertices
With reference to the figure above, H is the orthocenter of ΔABC
L, M, N are the feet of the perpendiculars from A,B,C respectively to the sides BC, CA, AB respectively.
In ΔALC, ∠ALC=90°, ∠LCA = C
⇒∠CAL = 90° – C
⇒∠HAM = 90° – C
In ΔABM, ∠MAB = A
In ΔAHM, AH = AM sec ∠HAE
Similarly,
The Orthocenter – Distance from Sides
With reference to the same figure above, ∠CAL = 90° – C
Similarly, ∠MBC = 90° – C
Also BH = 2R cos B
And in ΔHBL, HL = BH sin ∠HBL
Similarly,
Conclusion
We have obtained formulas for various lengths from the vertices and/or sides related to the circumcenter, centroid, incenter, and orthocenter. If we pay close attention, we can see some patterns emerge. In proper college textbook fashion, I leave the identification of these patterns to the reader! In the previous post, we had introduced the idea of an ex-center. We will deal with various formulas related to the ex-centers and the triangle formed by the ex-centers in the next post. We will also study what is known as the pedal triangle.
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