Triangles — CDS Elementary Mathematics

Formulas (3)

• Angle sum and the exterior angle · Angle sum and exterior angle
  $$A + B + C = 180^\circ, \qquad \angle ACD = A + B$$
• Where two angle bisectors meet · Bisector angles
  $$\angle BIC = 90^\circ + \tfrac{A}{2}, \qquad \angle BEC = 90^\circ - \tfrac{A}{2}$$
• Isosceles triangles and parallel lines · Base angles of an isosceles triangle
  $$AB = AC \;\Rightarrow\; \angle B = \angle C = 90^\circ - \tfrac{A}{2}$$

Watch out for (4)

• Opposite angles, not the adjacent one
  → Angle sum and the exterior angle
• Some questions answer themselves
  → Angle sum and the exterior angle
• Two angles form where the bisectors cross
  → Where two angle bisectors meet
• Equal angles face the equal sides
  → Isosceles triangles and parallel lines

Formulas (3)

• The triangle inequality · Triangle inequality
  $$|b - c| < a < b + c$$
• The larger side faces the larger angle · Acute, right or obtuse
  $$c^2 \lessgtr a^2 + b^2 \quad (c \text{ the longest side})$$
• The medians against the perimeter · Sum of the medians
  $$\tfrac34 (a + b + c) < m_a + m_b + m_c < a + b + c$$

Watch out for (4)

• Equal is not enough
  → The triangle inequality
• When two options are both true
  → The triangle inequality
• Compare the angles, not the picture
  → The larger side faces the larger angle
• Read which way the inequality points
  → The medians against the perimeter

Formulas (3)

• The congruence rules · Rules that fix a triangle
  $$\text{SSS},\ \text{SAS},\ \text{ASA},\ \text{AAS},\ \text{RHS} \qquad (\text{not AAA, not SSA})$$
• Similar triangles and the shared-angle pattern · Shared-angle similarity
  $$\angle ADC = \angle BAC \;\Rightarrow\; AC^2 = DC \cdot BC$$
• Crossing lines between two poles · Height of the crossing point
  $$h = \dfrac{ab}{a + b}$$

Watch out for (3)

• The angle must be the included one
  → The congruence rules
• Match vertices by the statement, not by the letters' positions
  → Similar triangles and the shared-angle pattern
• Not the average
  → Crossing lines between two poles

Formulas (3)

• A line parallel to one side · Basic proportionality
  $$DE \parallel BC \;\Rightarrow\; \dfrac{AD}{DB} = \dfrac{AE}{EC}, \quad \dfrac{DE}{BC} = \dfrac{AD}{AB}$$
• The midpoint theorem · Midpoint theorem
  $$DE \parallel BC, \quad DE = \tfrac12 BC$$
• The angle-bisector theorem · Angle-bisector theorem
  $$\dfrac{BD}{DC} = \dfrac{AB}{AC}$$

Watch out for (3)

• DE over BC is part over WHOLE
  → A line parallel to one side
• Count every side once
  → The midpoint theorem
• Pair each segment with the side that touches it
  → The angle-bisector theorem

Formulas (3)

• Areas of similar triangles · Area ratio of similar triangles
  $$\dfrac{\text{Area}_1}{\text{Area}_2} = \left(\dfrac{\text{side}_1}{\text{side}_2}\right)^2$$
• Joining the midpoints, again and again · Repeated midpoint triangles
  $$\text{Area after } n \text{ steps} = \dfrac{\text{Area}}{4^{\,n}}$$
• Same height, so compare the bases · Common height
  $$\dfrac{[ABD]}{[ADC]} = \dfrac{BD}{DC}$$

Watch out for (3)

• The length ratio is the square root
  → Areas of similar triangles
• Count the steps, not the positions
  → Joining the midpoints, again and again
• The bases must lie on one line
  → Same height, so compare the bases

Formulas (3)

• Pythagorean triples · Triple generator
  $$(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$$
• Perimeter and area together · Hypotenuse from perimeter and area
  $$(a + b)^2 = c^2 + 4\Delta, \qquad c = \dfrac{P^2 - 4\Delta}{2P}$$
• Ladders, poles and walks · Pythagoras
  $$c^2 = a^2 + b^2$$

Watch out for (3)

• A ratio fixes only the shape
  → Pythagorean triples
• The squares of ALL three sides
  → Perimeter and area together
• Poles use the DIFFERENCE of heights
  → Ladders, poles and walks

Formulas (3)

• The altitude is leg × leg ÷ hypotenuse · Altitude to the hypotenuse
  $$p = \dfrac{ab}{c}, \qquad \dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}$$
• The three similar triangles · Geometric-mean relations
  $$p^2 = mn, \qquad AB^2 = m(m + n), \qquad AC^2 = n(m + n)$$
• Altitudes of any triangle · Altitude from the area
  $$h_a = \dfrac{2\Delta}{a}, \qquad a : b : c = \dfrac{1}{h_a} : \dfrac{1}{h_b} : \dfrac{1}{h_c}$$

Watch out for (3)

• Find the right angle first
  → The altitude is leg × leg ÷ hypotenuse
• A leg uses its own piece and the WHOLE hypotenuse
  → The three similar triangles
• Inverse, not direct
  → Altitudes of any triangle

Formulas (3)

• Apollonius' theorem · Apollonius' theorem
  $$AB^2 + AC^2 = 2\left(AD^2 + BD^2\right)$$
• Points on a leg of a right triangle · Two midpoints of the legs
  $$AQ^2 + CP^2 = \tfrac54\,AC^2$$
• Subtracting across an altitude · Across an altitude
  $$AB^2 - AC^2 = BD^2 - DC^2$$

Watch out for (3)

• Half the base, not the base
  → Apollonius' theorem
• Measure from the right angle
  → Points on a leg of a right triangle
• The general form needs a perpendicular
  → Subtracting across an altitude

Formulas (2)

• Circumradius and inradius · Right-triangle radii
  $$R = \dfrac c2, \qquad r = \dfrac{a + b - c}{2}$$
• Placing the triangle on axes · Circle in the corner of angle A
  $$\rho = r\,\dfrac{1 - \sin(A/2)}{1 + \sin(A/2)}$$

Reference tables (1)

Watch out for (3)

• 'On the triangle' is not 'inside'
  → The four centres and where they lie
• Halve it
  → Circumradius and inradius
• Half the angle
  → Placing the triangle on axes

Formulas (2)

• The sine rule · Sine rule
  $$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$$
• The cosine rule and ½ab sin C · Cosine rule and area
  $$a^2 = b^2 + c^2 - 2bc\cos A, \qquad \Delta = \tfrac12 bc\sin A$$

Watch out for (2)

• Sines, not angles
  → The sine rule
• cos 120° is negative
  → The cosine rule and ½ab sin C