User:Prophet of Why

For various proofs in mathematics 

Triangle Proof
Concurency

$$ \begin{array}{clcl} \overline{AB} & y=0 & \overline{AD} & y={a\over c+b}x \\ \overline{BC} & y={a\over c+b}x-{ab\over c-b} & \overline{BE} & y={a\over c-2b}x-{ab\over c-2b} \\ \overline{CA} & y={a\over c}x & \overline{CF} & y={2a\over 2c-b}x-{ab\over 2c-b} \end{array}$$

$$ \begin{array}{rcl} y & = & {a\over c+b}x \\ -[y & = & {a\over c- 2b}x-{ab\over c-2b}] \\ 0 & = & {ac-2ab-ac-ab\over (c+b)(c-2b)}x+{ab\over c-2b} \\ 0 & = & {-3ab\over (c+b)(c-2b)}x+{ab\over c-2b} \\ {-ab\over c-2b} & = & {-3ab\over (c+b)(c-2b)}x \\ 1 & = & {3\over c+b}x \\ {c+b\over 3} & = & x \\ y & = & {a\over c+b}x \\ y & = & \big({a\over c+b} \big) \big({c+b\over 3} \big) \\ y & = & {a\over 3} \end{array}$$

$$ \begin{array}{lccl} \overline{AD} & {a\over 3}& = & \big({a\over c+b}\big)\big({c+b\over 3}\big) \\ & 1 & = & 1 \\ \overline{BE} & {a\over 3} & = & {a{c+b\over 3}-ab\over c-2b}\\ & 1 & = & {c+b-3b\over c-2b} \\ & 1 & = & {c-2b\over c-2b} \\ & 1 & = & 1 \\ \overline{CF} & {a\over 3} & = & {2a{c+b\over 3}-ab\over 2c-b} \\ & 1 & = & {2c+2b-3b\over 2c-b} \\ & 1 & = & {2c-b\over 2c-b} \\ & 1 & = & 1 \end{array}$$

Thus, $$\overline{AD},\,\overline{BE},\,and\,\,\overline{CF}$$ pass through point $$\big(\tfrac{c+b}{3},\tfrac{a}{3}\big)$$, point $$G$$.

Area Sum

$$\begin{array}{lcllcl} \Delta AEG & = & \int_0^{c\over 2} {a\over c}x-{a\over c+b}x \, dx + \int_{c\over 2}^{c+b\over 3} {a\over c-2b}x-{ab\over c-2b}-{a\over c+b}x \, dx & \Delta AFG & = & \int_0^{b\over 2} {a\over c+b}x \, dx + \int_{b\over 2}^{c+b\over 3} {a\over c+b}x-\big({2a\over 2c-b}x-{ab\over 2c-b}\big) \, dx \\ \Delta BDG & = & \int_{c+b\over 3}^{c+b\over 2} {a\over c+b}x-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_{c+b\over 2}^b \big({a\over c-b}x-{ab\over c-b}\big)-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx & \Delta BFG & = & \int_{b\over 2}^{c+b\over 3} {2a\over 2c-b}x-{ab\over 2c-b} \, dx + \int_{c+b\over 3}^b {a\over c-2b}x-{ab\over c-2b} \, dx \\ \Delta CDG & = & \int_{c+b\over 3}^c \big({2a\over 2c-b}x-{ab\over 2c-b}\big)-{a\over c+b}x \, dx + \int_c^{c+b\over 2} \big({a\over c-b}x-{ab\over c-b}\big)-{a\over c+b}x \, dx & \Delta CEG & = & \int_{c\over 2}^{c+b\over 3} {a\over c}x-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_{c+b\over 3}^c {a\over c}x-\big({2a\over 2c-b}x-{ab\over 2c-b}\big) \, dx \\ \Delta ABC & = & \int_0^c {2a\over c+b}x \, dx + \int_c^b {a\over c-b}x-{ab\over c-b} \, dx \end{array}$$

$$begin{array}{rcl} \int_0^c {2a\over c+b}x \, dx + \int_c^b {a\over c-b}x-{ab\over c-b} \, dx & = & \int_0^{c\over 2} {a\over c}x-{a\over c+b}x \, dx + \int_{c\over 2}^{c+b\over 3} {a\over c-2b}x-{ab\over c-2b}-{a\over c+b}x \, dx + \int_0^{b\over 2} {a\over c+b}x \, dx + \int_{b\over 2}^{c+b\over 3} {a\over c+b}x-\big({2a\over 2c-b}x-{ab\over 2c-b}\big) \, dx + \int_{c+b\over 3}^{c+b\over 2} {a\over c+b}x-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_{c+b\over 2}^b \big({a\over c-b}x-{ab\over c-b}\big)-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_{b\over 2}^{c+b\over 3} {2a\over 2c-b}x-{ab\over 2c-b} \, dx + \int_{c+b\over 3}^b {a\over c-2b}x-{ab\over c-2b} \, dx + \int_{c+b\over 3}^c \big({2a\over 2c-b}x-{ab\over 2c-b}\big)-{a\over c+b}x \, dx + \int_c^{c+b\over 2} \big({a\over  c-b}x-{ab\over c-b}\big)-{a\over c+b}x \, dx + \int_{c\over 2}^{c+b\over 3} {a\over c}x-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_{c+b\over 3}^c {a\over c}x-\big({2a\over 2c-b}x-{ab\over 2c-b}\big) \, dx \\ & = & \int_0^{c\over 2} {a\over c}x-{a\over c+b}x \, dx + \int_{c\over 2}^{c+b\over 3} {a\over c+2b}x-{a\over c+b}x \, dx + \int_{c+b\over 3}^c {a\over c+b}-{a\over c+b}x \, dx + \int_0^{b\over 2} {a\over c+b}x \, dx + \int_{b\over 2}^{c+b\over 3} {a\over c+b}x \, dx + \int_{c+b\over 3}^c \big({2a\over 2c-b}x-{ab\over 2c-b}\big)-{a\over c+b}x \, dx + \int_{c+b\over 3}^c {a\over c+b}x-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_c^{c+b\over 2} {a\over c+b}x-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + \int_{c+b\over 2}^b \big({a\over c-b}x-{ab\over c-b}\big)-\big({a\over c-2b}x-{ab\over c-2b}\big) \, dx + int_{c+b\over 3}^c {a\over c-2b}x-{ab\over c-2b} \, dx + int_c^b {a\over c-2b}x-{ab\over c-2b} \, dx end{array}$$