feed back

HI
i use this circuit in LC meter
L1C1 are the tank circuit but where is the feedback for this L1c1 to make oscillator
its output is a sin wave but i don't know where its feed back

What feedback are you referring to?

(An odd circuit... ).

???

R3 provides positive feedback.

but
R3 provides DC feedback
feedback blocked by ac coupling

but c3 is block DC signal that is the feedback signal
so how feedback signal reach to LC oscillator ???

R3 makes the comparator work as a Schmitt trigger, possibly used to provide a nice square waveform for the processor?.

However, when the 'designer' can't even drive a relay correctly, I'm dubious about anything else he might have done

The comparator, together with R1/2/3/4/5 and C4, form a schmitt-trigger oscillator. Without C3 connected, it will oscillate at some low frequency (in the vicinity of 0.6/R2/C4 Hz). The voltage on C4 will oscillate between 1/3*Vcc and 2/3*Vcc, the comparator output will be either 0V or Vcc (roughly), and the comparator +ve input will be either 1/3*Vcc or 2/3*Vcc.

When you connect C3 and the LC resonant circuit, things change. When the comparator changes state, some charge will flow through C3 and will excite the LC tank. The LC tank will resonate and will trigger the comparator, which will in turn provide another charge through C3 (every half cycle) to keep the LC circuit oscillating. The frequency output by the oscillator is now determined (mostly) by the L and C values.

It's true that C3 blocks DC but it pass through AC. It's role is loss less AC coupling of LC tank to LM311 oscillator.
Reactance (AC resistance) of C3 at 1kHz is only 15.9 ohms which can be neglected.