Answer 1

`bar"PQ" = bar"PS" + bar"SQ"`

∴ `bar"a" = bar"c" - bar"d"`   ...(1)

`bar"PR" = bar"PS" + bar"SR"`

∴ `bar"b" = bar"c" + bar"d"`    ....(2)

Adding equations (1) and (2), we get

`bar"a" + bar"b" = (bar"c" - bar"d") + (bar"c" + bar"d") = 2bar"c"`

∴ `bar"c" = 1/2 (bar"a" + bar"b") = 1/2 (bar"a" + bar"b")`

`= (2bar"b" - (bar"a" + bar"b"))/2`

`= 1/2(bar"b" - bar"a") = 1/2bar"b" - 1/2bar"a"`

Hence, `bar"c" = 1/2 bar"a" + 1/2bar"b"` and `bar"d" = 1/2bar"b" - 1/2bar"a"`