Properties of Triangles

Properties of Triangles

(i) Area of the ΔABC:

The area of ΔABC (denoted by Δ or S) may be expressed in many ways as follows:

(i) Δ = 1/2 bc sin A = 1/2 ca sin B = 1/2 ab sin C.

(ii) Δ =

(iii) Δ =

(ii) Sine Rule

In any ΔABC,

(iii) Cosine Rule

In any ΔABC

(iv) Projection Rule

In any ΔABC,

a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A.

(v) Tangent Rule (Nepier’s Analogy)

In any ΔABC,

(vi) Trigonometrical Ratios of the Half-Angles of ΔABC

(a)

(b)

(c)

(vii) Formulae for Circum-radius R:

(a)

(b)

(viii) Formulae for In-radius r:

(a) r = Δ/s

(b) r = (s-a) tan A/2 = (s-b) tan B/2 = (s-c) tan C/2

(c)

(d) r = 4 R sin A/2 sin B/2 sin C/2

(ix) Formulae for ex-radii r ₁ , r ₂ , r ₃ :

(a)

(b) r ₁ = s tan A/2, r ₂ = s tan B/2, r ₃ = s tan C/2

(c)

(d) r ₁ = 4R sin A/2 cos B/2 cos C/2, r ₂ = 4R cos A/2 sin B/2 cos C/2,

r ₃ = 4R cos A/2 cos B/2 sin C/2.

(x) Orthocentre and Pedal triangle of any Triangle:

(xi) The distances of the orthocentre from the vertices and the sides:

(a) PA = 2R cos A, PB = 2R cos B, PC = 2R cos C

(b) PD = 2R cos B cos C, PE = 2R cos C cos A, PF = 2R cos A cos B.

(xii) Sides and Angles of the Pedal Triangle:

(i) EF = a cos A, DF = b cos B, DE = c cos C

(ii) ∠ EDF = 180° - 2A, ∠DEF = 180° - 2B, ∠EFD = 180° - 2C

(xiii) Length of the Medians:

If AD, BE & CF are the medians of the triangle ABC then

(xiv) Distance between the circumcentre and the Orthocentre:

If ‘O’ is the circumcentre and ‘P’ is the orthocentre then

OP = R

(xv) Distance between the Circumcentre and the Incentre:

If ‘O’ is the circumcentre and ‘I’ is the Incentre then

OI = R

(xvi) Ptolemy’s Theorem:

In a cyclic quadrilateral ABCD.

AC. BD = AB . CD + BC . AD.

(xvii) Cyclic Quadrilateral:

(xviii) Area of the Quadrilateral:

(xix) Regular Polygon:

(xx) Some More Relations Regarding a Triangle:

(a) a cos A + b cos B + c cos C = 4R sin A sin B sin C.

(b) a cot A + b cot B + c cot C = 2 (R+r)

(c) r ₁ + r ₂ + r ₃ = 4R + r.

(d) r ₁ r ₂ + r ₂ r ₃ + r ₃ r ₁ = s ² .

Related Topics

• Properties of Circles
• Inverse Circular Functions