Proof: Medians of Isosceles

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	distance XS = distance SY	if S is the midpoint of line XY, then distance XS = distance SY (Midpoint Property)
2	m∠XSY = 180	if S is the midpoint of line XY, then m∠XSY = 180 (Angle Formed by Midpoint)
3	(distance XS) + (distance SY) = distance XY	if m∠XSY = 180, then (distance XS) + (distance SY) = distance XY (Length is Sum of Parts)
4	(distance XS) + (distance XS) = distance XY	if (distance XS) + (distance SY) = distance XY and distance XS = distance SY, then (distance XS) + (distance XS) = distance XY (Substitute 2)
5	(distance XS) ⋅ 2 = distance XY	if (distance XS) + (distance XS) = distance XY, then (distance XS) ⋅ 2 = distance XY (double)
6	distance XS = (distance XY) ⋅ (1 / 2)	if (distance XS) ⋅ 2 = distance XY, then distance XS = (distance XY) ⋅ (1 / 2) (Multiply by 2)
7	distance XS = (distance XZ) ⋅ (1 / 2)	if distance XS = (distance XY) ⋅ (1 / 2) and distance XY = distance XZ, then distance XS = (distance XZ) ⋅ (1 / 2) (First Term Substitution)
8	distance XT = distance TZ	if T is the midpoint of line XZ, then distance XT = distance TZ (Midpoint Property)
9	m∠XTZ = 180	if T is the midpoint of line XZ, then m∠XTZ = 180 (Angle Formed by Midpoint)
10	(distance XT) + (distance TZ) = distance XZ	if m∠XTZ = 180, then (distance XT) + (distance TZ) = distance XZ (Length is Sum of Parts)
11	(distance XT) + (distance XT) = distance XZ	if (distance XT) + (distance TZ) = distance XZ and distance XT = distance TZ, then (distance XT) + (distance XT) = distance XZ (Substitute 2)
12	(distance XT) + (distance XT) = (distance XT) ⋅ 2	(distance XT) + (distance XT) = (distance XT) ⋅ 2 (addition_theorem)
13	(distance XT) ⋅ 2 = distance XZ	if (distance XT) + (distance XT) = (distance XT) ⋅ 2 and (distance XT) + (distance XT) = distance XZ, then (distance XT) ⋅ 2 = distance XZ (Transitive Property of Equality Variation 2)
14	distance XT = (distance XZ) ⋅ (1 / 2)	if (distance XT) ⋅ 2 = distance XZ, then distance XT = (distance XZ) ⋅ (1 / 2) (Multiply by 2)
15	distance XS = distance XT	if distance XS = (distance XZ) ⋅ (1 / 2) and distance XT = (distance XZ) ⋅ (1 / 2), then distance XS = distance XT (Transitive Property of Equality Variation 1)
16	distance XT = distance XS	if distance XS = distance XT, then distance XT = distance XS (Symmetric Property of Equality)
17	m∠YSX = 180	if m∠XSY = 180, then m∠YSX = 180 (Angle Symmetry Example 2)
18	m∠ZTX = 180	if m∠XTZ = 180, then m∠ZTX = 180 (Angle Symmetry Example 2)
19	m∠YXT = m∠ZXS	if m∠YSX = 180 and m∠ZTX = 180, then m∠YXT = m∠ZXS (Angle to Self)
20	distance YX = distance XZ	if distance XY = distance XZ, then distance YX = distance XZ (Distance Property 1)
21	distance YX = distance ZX	if distance YX = distance XZ, then distance YX = distance ZX (Distance Property 2)
22	△YXT ≅ △ZXS	if distance YX = distance ZX and m∠YXT = m∠ZXS and distance XT = distance XS, then △YXT ≅ △ZXS (SAS Property)
23	distance YT = distance ZS	if △YXT ≅ △ZXS, then distance YT = distance ZS (Congruent Triangles to Distance 3)

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